106 results on '"Giambattista Giacomin"'
Search Results
2. Localization, Big-Jump Regime and the Effect of Disorder for a Class of Generalized Pinning Models
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Giambattista Giacomin and Benjamin Havret
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Physics ,Generalization ,Probability (math.PR) ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,01 natural sciences ,Boltzmann distribution ,010305 fluids & plasmas ,Nonlinear system ,Delocalized electron ,Distribution (mathematics) ,60K35, 60K37, 82B44, 60K10 ,Heavy-tailed distribution ,0103 physical sciences ,FOS: Mathematics ,Statistical physics ,Renewal theory ,010306 general physics ,Random variable ,Mathematics - Probability ,Mathematical Physics - Abstract
One dimensional pinning models have been widely studied in the physical and mathematical literature, also in presence of disorder. Roughly speaking, they undergo a transition between a delocalized phase and a localized one. In mathematical terms these models are obtained by modifying the distribution of a discrete renewal process via a Boltzmann factor with an energy that contains only one body potentials. For some more complex models, notably pinning models based on higher dimensional renewals, it has been shown that other phases may be present. We study a generalization of the one dimensional pinning model in which the energy may depend in a nonlinear way on the contact fraction: this class of models contains the circular DNA case considered in the bio-physics literature. We give a full solution of this generalized pinning model in absence of disorder and show that another transition appears. In fact the systems may display up to three different regimes: delocalization, partial localization and full localization. What happens in the partially localized regime can be explained in terms of the "big-jump" phenomenon for sums of heavy tail random variables under conditioning. We then show that disorder completely smears this second transition and we are back to the delocalization versus localization scenario. In fact we show that the disorder, even if arbitrarily weak, is incompatible with the presence of a big-jump., 33 pages, 5 figures
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- 2020
3. Disorder and denaturation transition in the generalized Poland–Scheraga model
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Giambattista Giacomin, Quentin Berger, and Maha Khatib
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Physics ,Delocalized electron ,Phase transition ,Homogeneous ,Exponent ,Thermodynamics ,Ocean Engineering ,Denaturation (biochemistry) ,Smoothing - Abstract
We investigate the generalized Poland-Scheraga model, which is used in the bio-physical literature to model the DNA denaturation transition, in the case where the two strands are allowed to be non-complementary (and to have different lengths). The homogeneous model was recently studied from a mathematical point of view in [35, 7], via a 2-dimensional renewal approach, with a loop exponent 2+α (α > 0): it was found to undergo a localization/delocalization phase transition of order ν = min(1, α) −1 , together with – in general – other phase transitions. In this paper, we turn to the disordered model, and we address the question of the influence of disorder on the denaturation phase transition, that is whether adding an arbitrarily small amount of disorder (i.e. inhomogeneities) affects the critical properties of this transition. Our results are consistent with Harris' predictions for d-dimensional disordered systems (here d = 2). First, we prove that when α d/2), then disorder is irrelevant: the quenched and annealed critical points are equal, and the disordered denaturation phase transition is also of order ν = α −1. On the other hand, when α > 1, disorder is relevant: we prove that the quenched and annealed critical points differ. Moreover, we discuss a number of open problems, in particular the smoothing phenomenon that is expected to enter the game when disorder is relevant.
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- 2020
4. Random Polymer Models
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Giambattista Giacomin and Giambattista Giacomin
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- 2007
5. Phase Segregation Dynamics in Particle Systems with Long Range Interactions II: Interface Motion.
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Giambattista Giacomin and Joel Lebowitz
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- 1998
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6. The zeros of the partition function of the pinning model
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Giambattista Giacomin and Rafael L. Greenblatt
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Pinning models with complex potentials ,Probability (math.PR) ,Zeros of partition function ,Sharp asymptotic behavior of partition function ,Griffiths singularities ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Settore MAT/06 - Probabilita' e Statistica Matematica ,Settore MAT/07 ,FOS: Mathematics ,Geometry and Topology ,Settore MAT/07 - Fisica Matematica ,Mathematics - Probability ,Mathematical Physics ,82B27, 30C15, 31B05, 60E10, 82B44, 60K35 - Abstract
We aim at understanding for which (complex) values of the potential the pinning partition function vanishes. The pinning model is a Gibbs measure based on discrete renewal processes with power law inter-arrival distributions. We obtain some results for rather general inter-arrival laws, but we achieve a substantially more complete understanding for a specific one parameter family of inter-arrivals. We show, for such a specific family, that the zeros asymptotically lie on (and densely fill) a closed curve that, unsurprisingly, touches the real axis only in one point (the critical point of the model). We also perform a sharper analysis of the zeros close to the critical point and we exploit this analysis to approach the challenging problem of Griffiths singularities for the disordered pinning model. The techniques we exploit are both probabilistic and analytical. Regarding the first, a central role is played by limit theorems for heavy tail random variables. As for the second, potential theory and singularity analysis of generating functions, along with their interplay, will be at the heart of several of our arguments., Comment: 42 pages, 6 figures. Accepted for publication on "Mathematical Physics, Analysis and Geometry"
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- 2021
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7. Disorder and critical phenomena: the $$\alpha =0$$ α = 0 copolymer model
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Quentin Berger, Giambattista Giacomin, and Hubert Lacoin
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Statistics and Probability ,Phase transition ,Critical phenomena ,010102 general mathematics ,Order (ring theory) ,Renormalization group ,Critical value ,Condensed Matter::Disordered Systems and Neural Networks ,01 natural sciences ,Renormalization ,010104 statistics & probability ,Delocalized electron ,Exponent ,0101 mathematics ,Statistics, Probability and Uncertainty ,Analysis ,Mathematical physics ,Mathematics - Abstract
The copolymer model is a disordered system built on a discrete renewal process with inter-arrival distribution that decays in a regularly varying fashion with exponent $$1+ \alpha \;\geqslant \;1$$ . It exhibits a localization transition which can be characterized in terms of the free energy of the model: the free energy is zero in the delocalized phase and it is positive in the localized phase. This transition, which is observed when tuning the mean h of the disorder variable, has been tackled in the physics literature notably via a renormalization group procedure that goes under the name of strong disorder renormalization. We focus on the case $$\alpha =0$$ —the critical value $$h_c(\beta )$$ of the parameter h is exactly known (for every strength $$\beta $$ of the disorder) in this case—and we provide precise estimates on the critical behavior. Our results confirm the strong disorder renormalization group prediction that the transition is of infinite order, namely that when $$h\searrow h_c(\beta )$$ the free energy vanishes faster than any power of $$h-h_c(\beta )$$ . But we show that the free energy vanishes much faster than the physicists’ prediction.
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- 2018
8. Singular Behavior of the Leading Lyapunov Exponent of a Product of Random $${2 \times 2}$$ 2 × 2 Matrices
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Rafael L. Greenblatt, Giuseppe Genovese, and Giambattista Giacomin
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Physics ,Pure mathematics ,010102 general mathematics ,One-dimensional space ,Statistical and Nonlinear Physics ,Infinite product ,Lyapunov exponent ,01 natural sciences ,symbols.namesake ,Distribution (mathematics) ,Product (mathematics) ,0103 physical sciences ,symbols ,0101 mathematics ,Invariant (mathematics) ,010306 general physics ,Random variable ,Mathematical Physics ,Probability measure - Abstract
We consider a certain infinite product of random \({2 \times 2}\) matrices appearing in the solution of some 1 and 1 + 1 dimensional disordered models in statistical mechanics, which depends on a parameter \({\varepsilon > 0}\) and on a real random variable with distribution \({\mu}\). For a large class of \({\mu}\), we prove the prediction by Derrida and Hilhorst (J Phys A 16:2641, 1983) that the Lyapunov exponent behaves like \({C \epsilon^{2 \alpha}}\) in the limit \({\epsilon \searrow 0}\), where \({\alpha \in (0,1)}\) and \({C > 0}\) are determined by \({\mu}\). Derrida and Hilhorst performed a two-scale analysis of the integral equation for the invariant distribution of the Markov chain associated to the matrix product and obtained a probability measure that is expected to be close to the invariant one for small \({\epsilon}\). We introduce suitable norms and exploit contractivity properties to show that such a probability measure is indeed close to the invariant one in a sense that implies a suitable control of the Lyapunov exponent.
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- 2017
9. Stochastic Dynamics Out of Equilibrium : Institut Henri Poincaré, Paris, France, 2017
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Giambattista Giacomin, Stefano Olla, Ellen Saada, Herbert Spohn, Gabriel Stoltz, Giambattista Giacomin, Stefano Olla, Ellen Saada, Herbert Spohn, and Gabriel Stoltz
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- Probabilities, Differential equations
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Stemming from the IHP trimester'Stochastic Dynamics Out of Equilibrium', this collection of contributions focuses on aspects of nonequilibrium dynamics and its ongoing developments.It is common practice in statistical mechanics to use models of large interacting assemblies governed by stochastic dynamics. In this context'equilibrium'is understood as stochastically (time) reversible dynamics with respect to a prescribed Gibbs measure. Nonequilibrium dynamics correspond on the other hand to irreversible evolutions, where fluxes appear in physical systems, and steady-state measures are unknown.The trimester, held at the Institut Henri Poincaré (IHP) in Paris from April to July 2017, comprised various events relating to three domains (i) transport in non-equilibrium statistical mechanics; (ii) the design of more efficient simulation methods; (iii) life sciences. It brought together physicists, mathematicians from many domains, computer scientists, as well as researchers working at the interface between biology, physics and mathematics.The present volume is indispensable reading for researchers and Ph.D. students working in such areas.
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- 2019
10. A Law of Large Numbers and Large Deviations for interacting diffusions on Erd\H{o}s-R\'enyi graphs
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Helge Dietert, Giambattista Giacomin, Fabio Coppini, Laboratoire de Probabilités, Statistique et Modélisation (LPSM (UMR_8001)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), ANR-11-IDEX-0005,USPC,Université Sorbonne Paris Cité(2011), ANR-15-CE40-0020,LSD,Modèles stochastiques en grande dimension pour la physique statistique hors équilibre(2015), European Project: 665850,H2020,H2020-MSCA-COFUND-2014,INSPIRE(2015), European Project: 609102,EC:FP7:PEOPLE,FP7-PEOPLE-2013-COFUND,PRESTIGE(2014), Laboratoire de Probabilités, Statistiques et Modélisations (LPSM (UMR_8001)), and ANR-11-IDEX-0005-02/11-IDEX-0005,USPC,USPC(2011)
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Particle system ,Class (set theory) ,Pure mathematics ,60K35, 82C20 ,Differential equation ,010102 general mathematics ,16. Peace & justice ,01 natural sciences ,Erdos-Renyi graphs ,Vlasov and McKean-Vlasov PDEs ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,010104 statistics & probability ,Mean field ,Mean field theory ,Interaction network ,Interacting diffusions ,Modeling and Simulation ,Large deviations theory ,0101 mathematics ,Realization (systems) ,Mathematics - Probability ,Mathematics ,A-law algorithm - Abstract
We consider a class of particle systems described by differential equations (both stochastic and deterministic), in which the interaction network is determined by the realization of an Erd\H{o}s-R\'enyi graph with parameter $p_n\in (0, 1]$, where $n$ is the size of the graph (i.e., the number of particles). If $p_n\equiv 1$ the graph is the complete graph (mean field model) and it is well known that, under suitable hypotheses, the empirical measure converges as $n\to \infty$ to the solution of a PDE: a McKean-Vlasov (or Fokker-Planck) equation in the stochastic case, or a Vlasov equation in the deterministic one. It has already been shown that this holds for rather general interaction networks, that include Erd\H{o}s-R\'enyi graphs with $\lim_n p_n n =\infty$, and properly rescaling the interaction to account for the dilution introduced by $p_n$. However, these results have been proven under strong assumptions on that initial datum which has to be chaotic, i.e. a sequence of independent identically distributed random variables. The aim of our contribution is to present results -- Law of Large Numbers and Large Deviation Principle -- assuming only the convergence of the empirical measure of the initial condition., Comment: 16 pages
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- 2018
11. Disorder and wetting transition: The pinned harmonic crystal in dimension three or larger
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Hubert Lacoin, Giambattista Giacomin, Laboratoire de Probabilités, Statistiques et Modélisations (LPSM (UMR_8001)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), and ANR-15-CE40-0020,LSD,Modèles stochastiques en grande dimension pour la physique statistique hors équilibre(2015)
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Statistics and Probability ,localization transition ,Logarithm ,82B44 ,FOS: Physical sciences ,critical behavior ,01 natural sciences ,Omega ,Lattice (order) ,60K35, 60K37, 82B27, 82B44 ,0103 physical sciences ,Gaussian free field ,FOS: Mathematics ,Lattice Gaussian free field ,Boundary value problem ,0101 mathematics ,ComputingMilieux_MISCELLANEOUS ,Mathematical Physics ,Mathematics ,Mathematical physics ,Random field ,Probability (math.PR) ,010102 general mathematics ,Mathematical Physics (math-ph) ,disordered pinning model ,disorder irrelevance ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,60K37 ,Wetting transition ,60K35 ,010307 mathematical physics ,Statistics, Probability and Uncertainty ,Random variable ,Mathematics - Probability ,82B27 - Abstract
We consider the Lattice Gaussian free field in $d+1$ dimensions, $d=3$ or larger, on a large box (linear size $N$) with boundary conditions zero. On this field two potentials are acting: one, that models the presence of a wall, penalizes the field when it enters the lower half space and one, the {\guillemotleft}pinning potential{\guillemotright} , that rewards visits to the proximity of the wall. The wall can be soft, i.e. the field has a finite penalty to enter the lower half plane, or hard when the penalty is infinite. In general the pinning potential is disordered and it gives on average a reward h in $\mathbb{R}$ (a negative reward is a penalty): the energetic contribution when the field at site x visits the pinning region is $\beta \omega_x+h$, $\{\omega_x\}_{x \in \mathbb{Z}^d}$ are IID centered and exponentially integrable random variables of unit variance and $\beta\ge 0$. In [E. Bolthausen, J.-D. Deuschel and O. Zeitouni, J. Math. Phys. 41 (2000), 1211-1223] it is shown that, when $\beta=0$ (that is, in the non disordered model), a delocalization-localization transition happens at $h=0$, in particular the free energy of the system is zero for $h \le 0$ and positive for $h>0$. We show that, for $\beta\neq 0$, the transition happens at $h=h_c(\beta):=- \log \mathbb{E} \exp(\beta \omega_x)$ and we find the precise asymptotic behavior of the logarithm of the free energy density of the system when $h \searrow h_c(\beta)$. In particular, we show that the transition is of infinite order in the sense that the free energy is smaller than any power of $h-h_c(\beta)$ in the neighborhood of the critical point and that disorder does not modify at all the nature of the transition. We also provide results on the behavior of the paths of the random field in the limit $N \to \infty$., Comment: 23 pages
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- 2018
12. A note on dynamical models on random graphs and Fokker-Planck equations
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Eric Luçon, Sylvain Delattre, Giambattista Giacomin, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC) - Université Paris Diderot - Paris 7 (UPD7) - Centre National de la Recherche Scientifique (CNRS), MAP5 - Mathématiques Appliquées à Paris 5 (MAP5), Centre National de la Recherche Scientifique (CNRS) - Institut National des Sciences Mathématiques et de leurs Interactions - Université Paris Descartes - Paris 5 (UPD5), Laboratoire de Probabilités et Modèles Aléatoires ( LPMA ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ), Mathématiques Appliquées à Paris 5 ( MAP5 - UMR 8145 ), Université Paris Descartes - Paris 5 ( UPD5 ) -Institut National des Sciences Mathématiques et de leurs Interactions-Centre National de la Recherche Scientifique ( CNRS ), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Fédération Parisienne de Modélisation Mathématique (FP2M), Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Université Pierre et Marie Curie - Paris 6 (UPMC), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), and Université Paris Descartes - Paris 5 (UPD5)-Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS)
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MSC: 82C20, 60K35 ,Pure mathematics ,mean field ,[MATH.MATH-PR] Mathematics [math]/Probability [math.PR] ,82C20, 60K35 ,FOS: Physical sciences ,01 natural sciences ,Fokker-Planck PDE ,010104 statistics & probability ,FOS: Mathematics ,nonlinear diffusion ,Limit (mathematics) ,0101 mathematics ,Diffusion (business) ,Mathematical Physics ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,Random graph ,Degree (graph theory) ,Horizon ,010102 general mathematics ,Probability (math.PR) ,Kuramoto models ,Complete graph ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,16. Peace & justice ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,interacting diffusions on graphs ,Mean field theory ,Fokker–Planck equation ,[ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR] ,Mathematics - Probability - Abstract
We address the issue of the proximity of interacting diffusion models on large graphs with a uniform degree property and a corresponding mean field model, i.e. a model on the complete graph with a suitably renormalized interaction parameter. Examples include Erd\H{o}s-R\'enyi graphs with edge probability $p_n$, $n$ is the number of vertices, such that $\lim_{n \to \infty}p_n n= \infty$. The purpose of this note it twofold: (1) to establish this proximity on finite time horizon, by exploiting the fact that both systems are accurately described by a Fokker-Planck PDE (or, equivalently, by a nonlinear diffusion process) in the $n=\infty$ limit; (2) to remark that in reality this result is unsatisfactory when it comes to applying it to systems with $N$ large but finite, for example the values of $N$ that can be reached in simulations or that correspond to the typical number of interacting units in a biological system., Comment: 14 pages, 1 figure. A few corrections, streamlined some explanations and improved Corollary 1.2
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- 2016
13. Oscillatory Critical Amplitudes in Hierarchical Models and the Harris Function of Branching Processes
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Ovidiu Costin, Giambattista Giacomin, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)
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Polynomial ,Logarithm ,010102 general mathematics ,Statistical and Nonlinear Physics ,Context (language use) ,Moment-generating function ,01 natural sciences ,Julia set ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,010104 statistics & probability ,Amplitude ,Bounded function ,Statistical physics ,0101 mathematics ,Random variable ,ComputingMilieux_MISCELLANEOUS ,Mathematical Physics ,Mathematics - Abstract
Oscillatory critical amplitudes have been repeatedly observed in hierarchical models and, in the cases that have been taken into consideration, these oscillations are so small to be hardly detectable. Hierarchical models are tightly related to iteration of maps and, in fact, very similar phenomena have been repeatedly reported in many fields of mathematics, like combinatorial evaluations and discrete branching processes. It is precisely in the context of branching processes with bounded off-spring that T. Harris, in 1948, first set forth the possibility that the logarithm of the moment generating function of the rescaled population size, in the super-critical regime, does not grow near infinity as a power, but it has an oscillatory prefactor. These oscillations have been observed numerically only much later and, while the origin is clearly tied to the discrete character of the iteration, the amplitude size is not so well understood. The purpose of this note is to reconsider the issue for hierarchical models and in what is arguably the most elementary setting -- the pinning model -- that actually just boils down to iteration of polynomial maps (and, notably, quadratic maps). In this note we show that the oscillatory critical amplitude for pinning models and the oscillating pre factor connected to the Harris random variable coincide. Moreover we make explicit the link between these oscillatory functions and the geometry of the Julia set of the map, making thus rigorous and quantitative some ideas set forth by B. Derrida, C. Itzykson and J. M. Luck (CMP 1984).
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- 2012
14. Landau damping in the Kuramoto model
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David Gérard-Varet, Giambattista Giacomin, Bastien Fernandez, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS), Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG), Université Paris Diderot - Paris 7 (UPD7), and Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Université Pierre et Marie Curie - Paris 6 (UPMC)
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Nuclear and High Energy Physics ,Perturbation (astronomy) ,FOS: Physical sciences ,35Q84, 82C44, 45D05, 35Q92, 92B25 ,Mathematical proof ,01 natural sciences ,Mathematics - Analysis of PDEs ,[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] ,0103 physical sciences ,FOS: Mathematics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Landau damping ,0101 mathematics ,010306 general physics ,[NLIN.NLIN-AO]Nonlinear Sciences [physics]/Adaptation and Self-Organizing Systems [nlin.AO] ,Mathematical Physics ,ComputingMilieux_MISCELLANEOUS ,Physics ,Kuramoto model ,010102 general mathematics ,Mathematical analysis ,Vlasov equation ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Nonlinear Sciences - Adaptation and Self-Organizing Systems ,Nonlinear system ,Adaptation and Self-Organizing Systems (nlin.AO) ,Stationary state ,Linear stability ,Analysis of PDEs (math.AP) - Abstract
We consider the Kuramoto model of globally coupled phase oscillators in its continuum limit, with individual frequencies drawn from a distribution with density of class $${C^n}$$ ( $${n\geq 4}$$ ). A criterion for linear stability of the uniform stationary state is established which, for basic examples in the literature, is equivalent to the standard condition on the coupling strength. We prove that, under this criterion, the Kuramoto order parameter, when evolved under the full nonlinear dynamics, asymptotically vanishes (with polynomial rate n) for every trajectory issued from a sufficiently small $${C^n}$$ perturbation. The proof uses techniques from the Analysis of PDEs and closely follows recent proofs of the nonlinear Landau damping in the Vlasov equation and Vlasov-HMF model.
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- 2015
15. DNA melting structures in the generalized Poland-Scheraga model
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Maha Khatib, Giambattista Giacomin, Quentin Berger, Laboratoire de Probabilités, Statistiques et Modélisations (LPSM (UMR_8001)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Université Libanaise, ANR-15-CE40-0020,LSD,Modèles stochastiques en grande dimension pour la physique statistique hors équilibre(2015), Laboratoire de Probabilités, Statistique et Modélisation (LPSM (UMR_8001)), and ANR-15-CE40-0020,LSD,Large Stochastic Dynamical Models in Non-Equilibrium Statistical Physics(2015)
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Statistics and Probability ,Work (thermodynamics) ,Generalization ,01 natural sciences ,010104 statistics & probability ,Polymer Pinning Model ,Chain (algebraic topology) ,FOS: Mathematics ,Calculus ,Renewal theory ,Statistical physics ,0101 mathematics ,Representation (mathematics) ,Condensation phenomena ,Mathematics ,non-Cramér regime ,Quantitative Biology::Biomolecules ,Probability (math.PR) ,010102 general mathematics ,Biomolecules (q-bio.BM) ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Loop (topology) ,Sharp Deviation Estimates ,2010 MSC: 60K35, 82D60, 92C05, 60K05, 60F10 ,Quantitative Biology - Biomolecules ,DNA Denaturation ,FOS: Biological sciences ,Pairing ,Path (graph theory) ,Two-dimensional Renewal Processes ,Mathematics - Probability ,60K35, 82D60, 92C05, 60K05, 60F10 - Abstract
The Poland-Scheraga model for DNA denaturation, besides playing a central role in applications, has been widely studied in the physical and mathematical literature over the past decades. More recently a natural generalization has been introduced in the biophysics literature to overcome the limits of the original model, namely to allow an excess of bases -- i.e. a different length of the two single stranded DNA chains -- and to allow slippages in the chain pairing. The increased complexity of the model is reflected in the appearance of configurational transitions when the DNA is in double stranded form. In a previous work of two of the authors the generalized Poland-Scheraga model has been analyzed thanks to a representation in terms of a bivariate renewal process. In this work we exploit this representation farther and fully characterize the path properties of the system, making therefore explicit the geometric structures -- and the configurational transitions -- that are observed when the polymer is in the double stranded form. What we prove is that, when the excess of bases is not absorbed in a homogeneous fashion along the double stranded chain, then it either condensates in a single macroscopic loop or it accumulates into an unbound single strand free end., Comment: 29 pages, 5 figures
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- 2018
16. Noise, interaction, nonlinear dynamics and the origin of rhythmic behaviors
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Giambattista Giacomin, Christophe Poquet, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)
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Statistics and Probability ,rhythmic behaviors ,010102 general mathematics ,Large numbers ,Statistical mechanics ,Fokker–Planck PDE ,Coupled stochastic rotators ,coupled excitable systems ,01 natural sciences ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Nonlinear system ,Noise ,normally contracting manifold ,Character (mathematics) ,Control theory ,0103 physical sciences ,Spite ,Kuramoto synchronization model ,Statistical physics ,Limit (mathematics) ,0101 mathematics ,010306 general physics ,Set (psychology) ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
International audience; Large families of noisy interacting units (cells, individuals, components in a circuit, ...) exhibiting synchronization often exhibit oscillatory behaviors too. This is a well established empirical observation that has attracted a remarkable amount of attention, notably in life sciences, because of the central role played by internally generated rhythms. A certain number of elementary models that seem to capture the essence, or at least some essential features, of the phenomenon have been set forth, but the mathematical analysis is in any case very challenging and often out of reach. We focus on phase models, proposed and repeatedly considered by Y. Kuramoto and coauthors, and on the mathematical results that can be established. In spite of the fact that noise plays a crucial role, and in fact these models in abstract terms are just a special class of diffusions in high dimensional spaces, the core of the analysis is at the level of the PDE that provides an accurate description of the limit of a very large number of units in interaction. We will stress how the fundamental difficulty in dealing with these models is in their non-equilibrium character and the results we present for phase models are crucially related to the fact that, with a very special choice of the parameters, they reduce to an equilibrium statistical mechanics model.
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- 2015
17. Pinning and disorder relevance for the lattice Gaussian free field
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Hubert Lacoin and Giambattista Giacomin
- Subjects
Phase transition ,General Mathematics ,Gaussian ,FOS: Physical sciences ,01 natural sciences ,Upper and lower bounds ,Condensed Matter::Disordered Systems and Neural Networks ,010104 statistics & probability ,symbols.namesake ,Delocalized electron ,Quadratic equation ,Lattice (order) ,Gaussian free field ,60K35, 60K37, 82B27, 82B44 ,FOS: Mathematics ,0101 mathematics ,Mathematical Physics ,Mathematics ,Condensed matter physics ,Applied Mathematics ,010102 general mathematics ,Probability (math.PR) ,Mathematical Physics (math-ph) ,symbols ,Critical exponent ,Mathematics - Probability - Abstract
This paper provides a rigorous study of the localization transition for a Gaussian free field on $\mathbb{Z}^d$ interacting with a quenched disordered substrate that acts on the interface when the interface height is close to zero. The substrate has the tendency to localize or repel the interface at different sites and one can show that a localization-delocalization transition takes place when varying the average pinning potential $h$: the free energy density is zero in the delocalized regime, that is for $h$ smaller than a threshold $h_c$, and it is positive for $h>h_c$. For $d\ge 3$ we compute $h_c$ and we show that the transition happens at the same value as for the annealed model. However we can show that the critical behavior of the quenched model differs from the one of the annealed one. While the phase transition of the annealed model is of first order, we show that the quenched free energy is bounded above by $ (h-h_c)_+^2$ times a positive constant and that, for Gaussian disorder, the quadratic behavior is sharp. Therefore this provides an example in which a {\sl relevant disorder critical exponent} can be made explicit: in theoretical physics disorder is said to be {\sl relevant} when the disorder changes the critical behavior of a system and, while there are cases in which it is known that disorder is relevant, the exact critical behavior is typically unknown. For $d=2$ we are not able to decide whether the quenched and annealed critical points coincide, but we provide an upper bound for the difference between them., 55 pages, 1 figure. Added a statement on path properties, corrected misprints and reformulated some arguments
- Published
- 2015
- Full Text
- View/download PDF
18. Small noise and long time phase diffusion in stochastic limit cycle oscillators
- Author
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Assaf Shapira, Giambattista Giacomin, Christophe Poquet, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Probabilités, statistique, physique mathématique (PSPM), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), ANR-15-CE40-0020,LSD,Modèles stochastiques en grande dimension pour la physique statistique hors équilibre(2015), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Dephasing ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,Phase (waves) ,FOS: Physical sciences ,Dynamical Systems (math.DS) ,Quantitative Biology - Quantitative Methods ,01 natural sciences ,Noise (electronics) ,Small Noise Limit ,010305 fluids & plasmas ,Long Time Dynamics ,Stochastic differential equation ,[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] ,Limit cycle ,0103 physical sciences ,FOS: Mathematics ,Statistical physics ,Limit (mathematics) ,Mathematics - Dynamical Systems ,010306 general physics ,Stable Hyperbolic Limit Cycles ,Mathematical Physics ,Brownian motion ,Quantitative Methods (q-bio.QM) ,Mathematics ,Applied Mathematics ,Probability (math.PR) ,Mathematical Physics (math-ph) ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Stochastic Differential Equations ,Isochrons ,60H10, 34F05, 60F17, 82C31, 92B25 ,FOS: Biological sciences ,Brownian noise ,Mathematics - Probability ,Analysis - Abstract
We study the effect of additive Brownian noise on an ODE system that has a stable hyperbolic limit cycle, for initial data that are attracted to the limit cycle. The analysis is performed in the limit of small noise - that is, we modulate the noise by a factor $\varepsilon \searrow 0$ - and on a long time horizon. We prove explicit estimates on the proximity of the noisy trajectory and the limit cycle up to times $\exp\left(c \varepsilon^{-2}\right)$, $c>0$, and we show both that on the time scale $\varepsilon^{-2}$ the "'dephasing" (i.e., the difference between noiseless and noisy system measured in a natural coordinate system that involves a phase) is close to a Brownian motion with constant drift, and that on longer time scales the dephasing dynamics is dominated, to leading order, by the drift. The natural choice of coordinates, that reduces the dynamics in a neighborhood of the cycle to a rotation, plays a central role and makes the connection with the applied science literature in which noisy limit cycle dynamics are often reduced to a diffusion model for the phase of the limit cycle., 24 pages, 1 figure. Small changes, added four references
- Published
- 2015
- Full Text
- View/download PDF
19. Weak noise and non-hyperbolic unstable fixed points: sharp estimates on transit and exit times
- Author
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Mathieu Merle, Giambattista Giacomin, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Statistics and Probability ,Probability (math.PR) ,Mathematical analysis ,WKB analysis ,Schrödinger equation ,Fixed point ,Fixed-point property ,Power law ,stochastic differential equations ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,martingale theory ,Stochastic differential equation ,symbols.namesake ,Mathematics - Classical Analysis and ODEs ,unstable non-hyperbolic fixed points ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,Limit (mathematics) ,Random variable ,Mathematics - Probability ,Brownian motion ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
We consider certain one dimensional ordinary stochastic differential equations driven by additive Brownian motion of variance $\varepsilon ^2$. When $\varepsilon =0$ such equations have an unstable non-hyperbolic fixed point and the drift near such a point has a power law behavior. For $\varepsilon >0$ small, the fixed point property disappears, but it is replaced by a random escape or transit time which diverges as $\varepsilon \searrow0$. We show that this random time, under suitable (easily guessed) rescaling, converges to a limit random variable that essentially depends only on the power exponent associated to the fixed point. Such random variables, or laws, have therefore a universal character and they arise of course in a variety of contexts. We then obtain quantitative sharp estimates, notably tail properties, on these universal laws., Published at http://dx.doi.org/10.3150/14-BEJ643 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)
- Published
- 2015
20. A Numerical Approach to Copolymers at Selective Interfaces
- Author
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Francesco Caravenna, Massimiliano Gubinelli, Giambattista Giacomin, Benassù, Serena, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Dipartimento di Matematica Applicata [Pisa] (DMA), Giacomin, Giambattista, Caravenna, F, Giacomin, G, and Gubinelli, M
- Subjects
[MATH.MATH-PR] Mathematics [math]/Probability [math.PR] ,Phase transition ,Large Deviation ,Phase (waves) ,FOS: Physical sciences ,Transfer Matrix Approach ,Disordered Model ,01 natural sciences ,Upper and lower bounds ,Statistical Tests ,010104 statistics & probability ,Delocalized electron ,Copolymer ,[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST] ,Disordered Models ,[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] ,Critical line ,0103 physical sciences ,FOS: Mathematics ,Concentration of Measure ,MSC 2000: 60K37, 82B44, 82B80 ,Statistical physics ,[MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph] ,0101 mathematics ,010306 general physics ,[MATH.MATH-ST] Mathematics [math]/Statistics [math.ST] ,Scaling ,Mathematical Physics ,Brownian motion ,Physics ,Corrections to Laplace estimate ,Localization Transition ,Large Deviations ,Copolymers ,Probability (math.PR) ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Partition function (mathematics) ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Corrections to Laplace estimates ,Mathematics - Probability - Abstract
We consider a model of a random copolymer at a selective interface which undergoes a localization/delocalization transition. In spite of the several rigorous results available for this model, the theoretical characterization of the phase transition has remained elusive and there is still no agreement about several important issues, for example the behavior of the polymer near the phase transition line. From a rigorous viewpoint non coinciding upper and lower bounds on the critical line are known. In this paper we combine numerical computations with rigorous arguments to get to a better understanding of the phase diagram. Our main results include: - Various numerical observations that suggest that the critical line lies strictly in between the two bounds. - A rigorous statistical test based on concentration inequalities and super-additivity, for determining whether a given point of the phase diagram is in the localized phase. This is applied in particular to show that, with a very low level of error, the lower bound does not coincide with the critical line. - An analysis of the precise asymptotic behavior of the partition function in the delocalized phase, with particular attention to the effect of rare atypical stretches in the disorder sequence and on whether or not in the delocalized regime the polymer path has a Brownian scaling. - A new proof of the lower bound on the critical line. This proof relies on a characterization of the localized regime which is more appealing for interpreting the numerical data., accepted for publication on J. Stat. Phys
- Published
- 2006
21. Scaling limits of equilibrium wetting models in (1+1)–dimension
- Author
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Jean Dominique Deuschel, Lorenzo Zambotti, Giambattista Giacomin, Benassù, Serena, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Statistics and Probability ,[MATH.MATH-PR] Mathematics [math]/Probability [math.PR] ,010102 general mathematics ,Boundary (topology) ,Geometry ,Random walk ,01 natural sciences ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,010104 statistics & probability ,Scaling limit ,Probability theory ,Wetting transition ,Upper half-plane ,Statistical physics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Scaling ,Analysis ,Brownian motion ,Mathematics - Abstract
We study the path properties for the δ-pinning wetting model in (1+1)–dimension. In other terms, we consider a random walk model with fairly general continuous increments conditioned to stay in the upper half plane and with a δ-measure reward for touching zero, that is the boundary of the forbidden region. It is well known that such a model displays a localization/delocalization transition, according to the size of the reward. Our focus is on getting a precise pathwise description of the system, in both the delocalized phase, that includes the critical case, and in the localized one. From this we extract the (Brownian) scaling limits of the model.
- Published
- 2004
22. On stochastic domination in the BrascampLieb framework
- Author
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Giambattista Giacomin
- Subjects
Continuous-time stochastic process ,Pure mathematics ,General Mathematics ,Gaussian ,Stochastic dominance ,Function (mathematics) ,Combinatorics ,Moment (mathematics) ,symbols.namesake ,symbols ,Stochastic optimization ,Random variable ,Complement (set theory) ,Mathematics - Abstract
We exploit a recent approach to Brascamp–Lieb inequalities, due to Caffarelli [ 5 ], and reconsider earlier approaches to establish stochastic domination inequalities between Gaussian variables and random variables with density of the form $g\cdot h, g$ a Gaussian density and $h$ a log-concave or log-convex function. These extend to inequalities on random vectors via a classical result by Prekopa and Leindler and they complement the Brascamp–Lieb moment inequalities. Some applications to a class of Gibbs measures, the anharmonic crystals , are developed.
- Published
- 2003
23. Enhanced interface repulsion from quenched hard–wall randomness
- Author
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Daniela Bertacchi, Giambattista Giacomin, Bertacchi, D, and Giacomin, G
- Subjects
Statistics and Probability ,large deviation ,Field (physics) ,Gaussian ,FOS: Physical sciences ,Geometry ,82B24 ,60K35 ,60G15 ,quenched and annealed model ,harmonic crystal ,Upper and lower bounds ,rough substrate ,symbols.namesake ,entropic repulsion ,Probability theory ,FOS: Mathematics ,extrema of random field ,Mathematical Physics ,Randomness ,Mathematics ,Gaussian field ,Random field ,Probability (math.PR) ,Mathematical analysis ,random walks ,Mathematical Physics (math-ph) ,Random walk ,MAT/06 - PROBABILITA E STATISTICA MATEMATICA ,Bounded function ,symbols ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,Analysis - Abstract
We consider the harmonic crystal on the d-dimensional lattice, d larger or equal to 3, that is the centered Gaussian field $\phi$ with covariance given by the Green function of the simple random walk on $Z^d$. Our main aim is to obtain quantitative information on the repulsion phenomenon that arises when we condition the field to be larger than an IID field $\sigma$ (which is also independent of $\phi$), for every x in a large region $D_N=ND\cap \Z^d$, with N a positive integer and $D \subset\R^d$. We are mostly motivated by results for given typical realizations of the $\sigma$ (quenched set-up), since the conditioned harmonic crystal may be seen as a model for an equilibrium interface, constrained not to go below a inhomogeneous substrate that acts as a hard wall. We consider various types of substrate and we observe that the interface is pushed away from the wall much more than in the case of a flat wall as soon as the upward tail of $\sigma$ is heavier than Gaussian, while essentially no effect is observed if the tail is sub--Gaussian. In the critical case, that is the one of approximately Gaussian tail, the interplay of the two sources of randomness, $\phi$ and $\sigma$, leads to an enhanced repulsion effect of additive type., Comment: 28 pages
- Published
- 2002
24. Noise-induced Periodicity: Some Stochastic Models for Complex Biological SystemsMathematical Models and Methods for Planet Earth
- Author
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DAI PRA, Paolo, Giambattista, Giacomin, and Daniele, Regoli
- Published
- 2014
25. Coherence Stability and Effect of Random Natural Frequencies in Populations of Coupled Oscillators
- Author
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Christophe Poquet, Eric Luçon, Giambattista Giacomin, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Université Pierre et Marie Curie - Paris 6 (UPMC)
- Subjects
Population ,FOS: Physical sciences ,01 natural sciences ,Stability (probability) ,010104 statistics & probability ,Limit (mathematics) ,0101 mathematics ,education ,Brownian motion ,Mathematical Physics ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,education.field_of_study ,Partial differential equation ,Kuramoto model ,010102 general mathematics ,Mathematical analysis ,Hyperbolic manifold ,Mathematical Physics (math-ph) ,Nonlinear Sciences - Adaptation and Self-Organizing Systems ,3. Good health ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,37N25, 82C26, 82C31, 92B25 ,Ordinary differential equation ,Quantitative Biology - Neurons and Cognition ,FOS: Biological sciences ,Neurons and Cognition (q-bio.NC) ,Adaptation and Self-Organizing Systems (nlin.AO) ,Analysis - Abstract
We consider the (noisy) Kuramoto model, that is a population of N oscillators, or rotators, with mean-field interaction. Each oscillator has its own randomly chosen natural frequency (quenched disorder) and it is stirred by Brownian motion. In the limit N goes to infty this model is accurately described by a (deterministic) Fokker-Planck equation. We study this equation and obtain quantitatively sharp results in the limit of weak disorder. We show that, in general, even when the natural frequencies have zero mean the oscillators synchronize (for sufficiently strong interaction) around a common rotating phase, whose frequency is sharply estimated. We also establish the stability properties of these solutions (in fact, limit cycles). These results are obtained by identifying the stable hyperbolic manifold of stationary solutions of an associated non disordered model and by exploiting the robustness of hyperbolic structures under suitable perturbations. When the disorder distribution is symmetric the speed vanishes and there is a one parameter family of stationary solutions : in this case we provide more precise stability estimates. The methods we use apply beyond the Kuramoto model and we develop here the case of active rotator models, that is the case in which the dynamics of each rotator in absence of interaction and noise is not simply a rotation., Comment: 33 pages, 3 figures
- Published
- 2014
26. Noise-induced periodicity: some stochastic models for complex biological systems
- Author
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Daniele Regoli, Giambattista Giacomin, Paolo Dai Pra, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), A. Celletti, U. Locatelli, T. Ruggeri, and E. Strickland
- Subjects
Hopf bifurcation ,Stylized fact ,Stochastic modelling ,Noise induced ,Computer science ,010102 general mathematics ,01 natural sciences ,010305 fluids & plasmas ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,symbols.namesake ,0103 physical sciences ,symbols ,Brownian noise ,Periodic orbits ,Statistical physics ,0101 mathematics ,ComputingMilieux_MISCELLANEOUS - Abstract
After a review of some examples of life science stochastic models, we propose a stylized model with characteristics inspired by the examples above, reproducing noise-induced pulsations as a collective macroscopic phenomenon.
- Published
- 2014
27. Log-periodic critical amplitudes : a perturbative approach
- Author
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Bernard Derrida, Giambattista Giacomin, Université Pierre et Marie Curie - Paris 6 (UPMC), Fédération de recherche du Département de physique de l'Ecole Normale Supérieure - ENS Paris (FRDPENS), École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Collège de France - Chaire Physique statistique, Collège de France (CdF (institution)), Laboratoire Polymères et Matériaux Avancés (LPMA), Institut de Chimie du CNRS (INC)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), and École normale supérieure - Paris (ENS Paris)
- Subjects
Large class ,Physics ,[PHYS]Physics [physics] ,Subdominant ,Monomial ,Statistical Mechanics (cond-mat.stat-mech) ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Scale invariance ,Renormalization ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Amplitude ,ComputingMilieux_MISCELLANEOUS ,Mathematical Physics ,Condensed Matter - Statistical Mechanics ,Mathematical physics - Abstract
Log-periodic amplitudes appear in the critical behavior of a large class of systems, in particular when a discrete scale invariance is present. Here we show how to compute these critical amplitudes perturbatively when they originate from a renormalization map which is close to a monomial. In this case, the log-periodic amplitudes of the subdominant corrections to the leading critical behavior can also be calculated., 17 pages, 2 figures
- Published
- 2014
28. [Untitled]
- Author
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Giambattista Giacomin, Thierry Bodineau, and Yvan Velenik
- Subjects
Physics ,Condensed matter physics ,010102 general mathematics ,Statistical and Nonlinear Physics ,01 natural sciences ,Reduction (complexity) ,010104 statistics & probability ,Quantum mechanics ,Degeneracy (biology) ,Ising model ,0101 mathematics ,Mathematics::Representation Theory ,Ground state ,Mathematical Physics - Abstract
We point out that there is no general relation between ground state degeneracy and finite-temperature fluctuations for tilted interfaces.
- Published
- 2001
29. Entropic repulsion for massless fields
- Author
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Jean-Dominique Deuschel and Giambattista Giacomin
- Subjects
Statistics and Probability ,Random walk in random environment ,Random field ,Applied Mathematics ,Massless fields ,Random walk ,Combinatorics ,Massless particle ,symbols.namesake ,Modeling and Simulation ,Modelling and Simulation ,symbols ,Even and odd functions ,Random surfaces ,Gibbs measure ,Convex function ,Jensen's inequality ,Laplace operator ,Gibbs measures ,Entropic repulsion ,Mathematics - Abstract
We consider the anharmonic crystal, or lattice massless field, with 0-boundary conditions outside D N =ND∩ Z d , D⊆ R d and N a large natural number, that is the finite volume Gibbs measure P N on {ϕ∈ R Z d :ϕ x =0 for every x∉DN} with Hamiltonian ∑ x∼y V(ϕ x −ϕ y ), V a strictly convex even function. We establish various bounds on P N (Ω + (D N )) , where Ω + (D N )={ϕ:ϕ x ⩾0 for all x∈DN}. Then we extract from these bounds the asymptotics (N→∞) of P N (·|Ω + (D N )) : roughly speaking we show that the field is repelled by a hard-wall to a height of O ( log N ) in d⩾3 and of O ( log N) in d=2. If we interpret ϕx as the height at x of an interface in a (d+1)-dimensional space, our results on the conditioned measure P N (·|Ω + (D N )) clarify some aspects of the effect of a hard-wall on an interface. Besides classical techniques, like the FKG inequalities and the Brascamp–Lieb inequalities for log-concave measures, we exploit a representation of the random field in term of a random walk in dynamical random environment (Helffer–Sjostrand representation).
- Published
- 2000
- Full Text
- View/download PDF
30. Large deviations and concentration properties for ∇ϕ interface models
- Author
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Dmitry Ioffe, Jean-Dominique Deuschel, and Giambattista Giacomin
- Subjects
Statistics and Probability ,Mathematical analysis ,Gibbs state ,Random walk ,Potential theory ,symbols.namesake ,symbols ,Large deviations theory ,Statistics, Probability and Uncertainty ,Gibbs measure ,Convex function ,Rate function ,Analysis ,Mathematics ,Second derivative - Abstract
We consider the massless field with zero boundary conditions outside D N ≡D∩ (ℤ d /N) (N∈ℤ+), D a suitable subset of ℝ d , i.e. the continuous spin Gibbs measure ℙ N on ℝ ℤd/N with Hamiltonian given by H(ϕ) = ∑ x,y:|x−y|=1 V(ϕ(x) −ϕ(y)) and ϕ(x) = 0 for x∈D N C . The interaction V is taken to be strictly convex and with bounded second derivative. This is a standard effective model for a (d + 1)-dimensional interface: ϕ represents the height of the interface over the base D N . Due to the choice of scaling of the base, we scale the height with the same factor by setting ξ N = ϕ/N. We study various concentration and relaxation properties of the family of random surfaces {ξ N } and of the induced family of gradient fields ∇ N ξ N as the discretization step 1/N tends to zero (N→∞). In particular, we prove a large deviation principle for {ξ N } and show that the corresponding rate function is given by ∫ D σ(∇u(x))dx, where σ is the surface tension of the model. This is a multidimensional version of the sample path large deviation principle. We use this result to study the concentration properties of ℙ N under the volume constraint, i.e. the constraint that (1/N d ) ∑ x∈DN ξ N (x) stays in a neighborhood of a fixed volume v > 0, and the hard–wall constraint, i.e. ξ N (x) ≥ 0 for all x. This is therefore a model for a droplet of volume v lying above a hard wall. We prove that under these constraints the field {ξ N of rescaled heights concentrates around the solution of a variational problem involving the surface tension, as it would be predicted by the phenomenological theory of phase boundaries. Our principal result, however, asserts local relaxation properties of the gradient field {∇ N ξ N (·)} to the corresponding extremal Gibbs states. Thus, our approach has little in common with traditional large deviation techniques and is closer in spirit to hydrodynamic limit type of arguments. The proofs have both probabilistic and analytic aspects. Essential analytic tools are ? p estimates for elliptic equations and the theory of Young measures. On the side of probability tools, a central role is played by the Helffer–Sjostrand [31] PDE representation for continuous spin systems which we rewrite in terms of random walk in random environment and by recent results of T. Funaki and H. Spohn [25] on the structure of gradient fields.
- Published
- 2000
31. Entropic Repulsion for the Free Field:¶Pathwise Characterization in d ≥ 3
- Author
-
Jean-Dominique Deuschel and Giambattista Giacomin
- Subjects
Conjecture ,Gaussian ,Mathematical analysis ,Complex system ,Statistical and Nonlinear Physics ,Covariance ,Free field ,Upper and lower bounds ,symbols.namesake ,Lattice (order) ,Quantum mechanics ,symbols ,Laplace operator ,Mathematical Physics ,Mathematics - Abstract
We study concentration properties of the lattice free field , i.e. the centered Gaussian field with covariance given by the Green function of the (discrete) Laplacian, when constrained to be positive in a region of volume O(N d ) (hard–wall condition). It has been shown in [3] that, as N→∞, the conditioned field is pushed to infinity: more precisely the typical value of the ϕ-variable to leading order is , and the exact value of c was found. It was moreover conjectured that the conditioned field, once this diverging height is subtracted, converges weakly to the lattice free field. Here we prove this conjecture, along with other explicit bounds, always in the direction of clarifying the intuitive idea that the free field with hard–wall conditioning merely translates away from the hard wall. We give also a proof, alternative to the one presented in [3], of the lower bound on the probability that the free field is everywhere positive in a region of volume N d .
- Published
- 1999
32. On the long time behavior of the stochastic heat equation
- Author
-
Lorenzo Bertini and Giambattista Giacomin
- Subjects
Statistics and Probability ,Partition function (statistical mechanics) ,Partial differential equation ,Mathematical finance ,Mathematical analysis ,Lyapunov exponent ,law.invention ,symbols.namesake ,Law of large numbers ,law ,Intermittency ,symbols ,Heat equation ,Statistics, Probability and Uncertainty ,Representation (mathematics) ,Analysis ,Mathematics - Abstract
We consider the stochastic heat equation in one space dimension and compute – for a particular choice of the initial datum – the exact long time asymptotic. In the Carmona-Molchanov approach to intermittence in non stationary random media this corresponds to the identification of the sample Lyapunov exponent. Equivalently, by interpreting the solution as the partition function of a directed polymer in a random environment, we obtain a weak law of large numbers for the quenched free energy. The result agrees with the one obtained in the physical literature via the replica method. The proof is based on a representation of the solution in terms of the weakly asymmetric exclusion process.
- Published
- 1999
33. The Sherrington-Kirkpatrick model with short range ferromagnetic interactions
- Author
-
Giambattista Giacomin, Francis Comets, and Joel L. Lebowitz
- Subjects
Ferromagnetism ,Condensed matter physics ,Ising spin ,Ising model ,General Medicine ,Mathematics ,Kirkpatrick model - Abstract
We study a model of Ising spins with short range ferromagnetic and long range SK interactions. We generalize the results obtained for the standard SK model, computing in particular the high temperature pressure.
- Published
- 1999
34. Exact Macroscopic Description of Phase Segregation in Model Alloys with Long Range Interactions
- Author
-
Giambattista Giacomin and Joel L. Lebowitz
- Subjects
Physics ,symbols.namesake ,Classical mechanics ,Lattice (order) ,Binary alloy ,Time evolution ,symbols ,General Physics and Astronomy ,Detailed balance ,Hamiltonian (quantum mechanics) ,Pair potential ,Mathematical physics - Abstract
We derive an exact nonlinear nonlocal macroscopic equation for the time evolution of the conserved order parameter $\ensuremath{\rho}(\mathbf{r},t)$ of a microscopic model binary alloy undergoing phase segregation: a d-dimensional lattice gas evolving via Kawasaki exchange dynamics, satisfying detailed balance for a Hamiltonian with a long range pair potential ${\ensuremath{\gamma}}^{d}J(\ensuremath{\gamma}|x|)$. The macroscopic evolution is on the spatial scale ${\ensuremath{\gamma}}^{\ensuremath{-}1}$ and time scale ${\ensuremath{\gamma}}^{\ensuremath{-}2}$, in the limit $\ensuremath{\gamma}\ensuremath{\rightarrow}0$. The domain coarsening, described by interface motion, is similar to that obtained from the Cahn-Hilliard equation.
- Published
- 1996
35. Disorder and Critical Phenomena Through Basic Probability Models : École D’Été De Probabilités De Saint-Flour XL – 2010
- Author
-
Giambattista Giacomin and Giambattista Giacomin
- Subjects
- Probabilities, Mathematics, System theory, Mathematical physics
- Abstract
Understanding the effect of disorder on critical phenomena is a central issue in statistical mechanics. In probabilistic terms: what happens if we perturb a system exhibiting a phase transition by introducing a random environment? The physics community has approached this very broad question by aiming at general criteria that tell whether or not the addition of disorder changes the critical properties of a model: some of the predictions are truly striking and mathematically challenging. We approach this domain of ideas by focusing on a specific class of models, the'pinning models,'for which a series of recent mathematical works has essentially put all the main predictions of the physics community on firm footing; in some cases, mathematicians have even gone beyond, settling a number of controversial issues. But the purpose of these notes, beyond treating the pinning models in full detail, is also to convey the gist, or at least the flavor, of the'overall picture,'which is, in many respects, unfamiliar territory for mathematicians.
- Published
- 2011
36. Synchronization and random long time dynamics for mean-field plane rotators
- Author
-
Lorenzo Bertini, Christophe Poquet, Giambattista Giacomin, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Statistics and Probability ,FOS: Physical sciences ,Measure (mathematics) ,kuramoto synchronization model ,finite size corrections to scaling limits ,Control theory ,Position (vector) ,FOS: Mathematics ,Statistical physics ,Langevin dynamics ,60K35, 37N25, 82C26, 82C31, 92B20 ,ComputingMilieux_MISCELLANEOUS ,Brownian motion ,Mathematical Physics ,Mathematics ,Probability measure ,Probability (math.PR) ,long time dynamics ,Mathematical Physics (math-ph) ,Statistical mechanics ,Empirical measure ,coupled rotators ,fokker-planck pde ,diffusion on stable invariant manifold ,Stable manifold ,Nonlinear Sciences - Adaptation and Self-Organizing Systems ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Statistics, Probability and Uncertainty ,Adaptation and Self-Organizing Systems (nlin.AO) ,Analysis ,Mathematics - Probability - Abstract
We consider the natural Langevin dynamics which is reversible with respect to the mean-field plane rotator (or classical spin XY) measure. It is well known that this model exhibits a phase transition at a critical value of the interaction strength parameter K, in the limit of the number N of rotators going to infinity. A Fokker-Planck PDE captures the evolution of the empirical measure of the system as N goes to infinity, at least for finite times and when the empirical measure of the system at time zero satisfies a law of large numbers. The phase transition is reflected in the fact that the PDE for K above the critical value has a stable manifold of stationary solutions, that are equivalent up to rotations. These stationary solutions are actually unimodal densities parametrized by the position of their maximum (the synchronization phase or center). We characterize the dynamics on times of order N and we show substantial deviations from the behavior of the solutions of the PDE. In fact, if the empirical measure at time zero converges as N goes to infinity to a probability measure (which is away from a thin set that we characterize) and if time is speeded up by N, the empirical measure reaches almost instantaneously a small neighborhood of the stable manifold, to which it then sticks and on which a non-trivial random dynamics takes place. In fact the synchronization center performs a Brownian motion with a diffusion coefficient that we compute. Our approach therefore provides, for one of the basic statistical mechanics systems with continuum symmetry, a detailed characterization of the macroscopic deviations from the large scale limit -- or law of large numbers -- due to finite size effects. But the interest for this model goes beyond statistical mechanics, since it plays a central role in a variety of scientific domains in which one aims at understanding synchronization phenomena., 47 pages, 1 figure
- Published
- 2012
37. Copolymers at Selective Interfaces: Settled Issues and Open Problems
- Author
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Fabio Lucio Toninelli, Francesco Caravenna, Giambattista Giacomin, Benassù, Serena, J.D. Deuschel, B. Gentz, W. König, U. Schmock, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), J.D. Deuschel, B. Gentz, W. König, U. Schmock, Deuschel, JD, Gentz, B, König, W, von Renesse, M, Scheutzow, M, Schmock, U, Caravenna, F, Giacomin, G, and Toninelli, F
- Subjects
Class (set theory) ,[MATH.MATH-PR] Mathematics [math]/Probability [math.PR] ,Theoretical computer science ,010102 general mathematics ,Probability (math.PR) ,Mathematical proof ,01 natural sciences ,Critical curve ,Field (computer science) ,Sketch ,Path property ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,010104 statistics & probability ,Mathematics Subject Classification ,Directed Polymers, Disorder, Localization, Copolymers at Selective Interfaces, Rare-Stretch Strategies, Fractional Moment Estimates ,60K35, 82B41, 82B44 ,FOS: Mathematics ,Point (geometry) ,0101 mathematics ,Algorithm ,Mathematics - Probability ,Mathematics - Abstract
We review the literature on the localization transition for the class of polymers with random potentials that goes under the name of copolymers near selective interfaces. We outline the results, sketch some of the proofs and point out the open problems in the field. We also present in detail some alternative proofs that simplify what one can find in the literature., In honor of Erwin Bolthausen on the occasion of his 65th birthday. 22 pages, 3 figures
- Published
- 2012
38. Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators
- Author
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Khashayar Pakdaman, Xavier Pellegrin, Giambattista Giacomin, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Equilibrium point ,37N25, 37B25, 92B25, 82C26 ,Applied Mathematics ,Kuramoto model ,010102 general mathematics ,General Physics and Astronomy ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Critical value ,01 natural sciences ,Nonlinear Sciences - Adaptation and Self-Organizing Systems ,010305 fluids & plasmas ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,0103 physical sciences ,Attractor ,Uniqueness ,Limit (mathematics) ,Statistical physics ,0101 mathematics ,Invariant (mathematics) ,Adaptation and Self-Organizing Systems (nlin.AO) ,Saddle ,Mathematical Physics ,Mathematics - Abstract
We study the dynamics of the large N limit of the Kuramoto model of coupled phase oscillators, subject to white noise. We introduce the notion of shadow inertial manifold and we prove their existence for this model, supporting the fact that the long term dynamics of this model is finite dimensional. Following this, we prove that the global attractor of this model takes one of two forms. When coupling strength is below a critical value, the global attractor is a single equilibrium point corresponding to an incoherent state. Conversely, when coupling strength is beyond this critical value, the global attractor is a two-dimensional disk composed of radial trajectories connecting a saddle equilibrium (the incoherent state) to an invariant closed curve of locally stable equilibria (partially synchronized state). Our analysis hinges, on the one hand, upon sharp existence and uniqueness results and their consequence for the existence of a global attractor, and, on the other hand, on the study of the dynamics in the vicinity of the incoherent and synchronized equilibria. We prove in particular non-linear stability of each synchronized equilibrium, and normal hyperbolicity of the set of such equilibria. We explore mathematically and numerically several properties of the global attractor, in particular we discuss the limit of this attractor as noise intensity decreases to zero., revised version, 28 pages, 4 figures
- Published
- 2011
39. Disorder relevance at marginality and critical point shift
- Author
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Fabio Lucio Toninelli, Hubert Lacoin, Giambattista Giacomin, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Physique de l'ENS Lyon (Phys-ENS), École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), ANR-05-BLAN-0322,POLINTBIO,Polymères, Interfaces et Systèmes Désordonnés : entre Mathématiques, Physique et Biologie(2005), École normale supérieure - Lyon (ENS Lyon)-Université Claude Bernard Lyon 1 (UCBL), ANR: POLINTBIO,POLINTBIO, Giacomin, Giambattista, and Programme non thématique - Appel à projets de recherche - Polymères, Interfaces et Systèmes Désordonnés : entre Mathématiques, Physique et Biologie - - POLINTBIO2005 - ANR-05-BLAN-0322 - JCJC - VALID
- Subjects
Many-body interactions ,Statistics and Probability ,82B44 ,[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] ,FOS: Physical sciences ,Harris criterion ,Geometry ,01 natural sciences ,010104 statistics & probability ,[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] ,[MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph] ,0101 mathematics ,Mathematical Physics ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,60K35 ,82B27 ,60K37 ,010102 general mathematics ,Mathematical Physics (math-ph) ,Marginal disorder ,[PHYS.MPHY] Physics [physics]/Mathematical Physics [math-ph] ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Disordered pinning models ,Statistics, Probability and Uncertainty ,Humanities - Abstract
Recently the renormalization group predictions on the effect of disorder on pinning models have been put on mathematical grounds. The picture is particularly complete if the disorder is 'relevant' or 'irrelevant' in the Harris criterion sense: the question addressed is whether quenched disorder leads to a critical behavior which is different from the one observed in the pure, i.e. annealed, system. The Harris criterion prediction is based on the sign of the specific heat exponent of the pure system, but it yields no prediction in the case of vanishing exponent. This case is called 'marginal', and the physical literature is divided on what one should observe for marginal disorder, notably there is no agreement on whether a small amount of disorder leads or not to a difference between the critical point of the quenched system and the one for the pure system. In a previous work (arXiv:0811.0723) we have proven that the two critical points differ at marginality of at least exp(-c/beta^4), where c>0 and beta^2 is the disorder variance, for beta in (0,1) and Gaussian IID disorder. The purpose of this paper is to improve such a result: we establish in particular that the exp(-c/beta^4) lower bound on the shift can be replaced by exp(-c(b)/beta^b), c(b)>0 for b>2 (b=2 is the known upper bound and it is the result claimed in [Derrida, Hakim, Vannimenus, JSP 1992]), and we deal with very general distribution of the IID disorder variables. The proof relies on coarse graining estimates and on a fractional moment-change of measure argument based on multi-body potential modifications of the law of the disorder., 30 pages
- Published
- 2011
40. The Coarse Graining Procedure
- Author
-
Giambattista Giacomin
- Subjects
Moment (mathematics) ,Arbitrarily large ,Partition function (quantum field theory) ,Homogeneous ,Granularity ,Statistical physics ,Mathematics - Abstract
This chapter develops in detail the most advanced of the two coarse graining techniques employed in the previous chapter. Roughly, it consists in looking at the system in blocks of finite size k, which essentially is the annealed correlation length: if we can get suitable estimates for systems up to that size k, we can bound the fractional moment of the partition function of the (arbitrarily large) system in terms of the partition function of a homogeneous model with pinning parameter that depend on the estimates up to size k.
- Published
- 2011
41. Homogeneous Pinning Systems: A Class of Exactly Solved Models
- Author
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Giambattista Giacomin
- Subjects
Physics ,Class (set theory) ,Phase transition ,Markov chain ,Geometry ,Ising model ,Statistical mechanics ,Renewal theory ,Statistical physics ,Partition function (mathematics) ,Random walk - Abstract
We introduce a class of statistical mechanics non-disordered models – the homogeneous pinning models – starting with the particular case of random walk pinning. We solve the model in the sense that we compute the precise asymptotic behavior of the partition function of the model. In particular, we obtain a formula for the free energy and show that the model exhibits a phase transition, in fact a localization/delocalization transition. We focus in particular on the critical behavior, that is on the behavior of the system close to the phase transition. The approach is then generalized to a general class of Markov chain pinning, which is more naturally introduced in terms of (discrete) renewal processes. We complete the chapter by introducing the crucial notion of correlation length and by giving an overview of the applications of pinning models. Ising models are presented at this stage because pinning systems appear naturally as limits of two dimensional Ising models with suitably chosen interaction potentials. In spite of the fact that these lecture notes may be read focusing exclusively on pinning, the physical literature on disordered systems and Ising models cannot be easily disentangled. So a full appreciation of some physical arguments/discussions in these notes does require being acquainted with Ising models.
- Published
- 2011
42. Path Properties
- Author
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Giambattista Giacomin
- Published
- 2011
43. Irrelevant Disorder Estimates
- Author
-
Giambattista Giacomin
- Subjects
Discrete mathematics ,Heuristic ,Exponent ,InformationSystems_MISCELLANEOUS ,Mathematics - Abstract
We introduce the Harris criterion and we provide two heuristic arguments in favor of this criterion. In particular we introduce the notion of relevant and irrelevant disorder. We then prove that disorder is irrelevant when the inter-arrival exponent α is smaller than 1 ∕ 2 and β is not too large.
- Published
- 2011
44. Relevant Disorder Estimates: The Smoothing Phenomenon
- Author
-
Giambattista Giacomin
- Subjects
Physics ,Phase transition ,Homogeneous ,Phenomenon ,Econometrics ,Ising model ,Statistical physics ,Heuristic argument ,Condensed Matter::Disordered Systems and Neural Networks ,Critical exponent ,Smoothing - Abstract
We show that, for α > 1 ∕ 2 and as soon as β > 0, disorder is relevant, in the sense that the critical behavior of the disordered system differs from the one of the pure, i.e. homogeneous, system. We do this by establishing a smoothing inequality for the free energy. We then review the literature on the effect of the disorder on phase transitions. In doing so we will present a number of physical predictions on disordered Ising models that are challenges for mathematicians.
- Published
- 2011
45. Introduction
- Author
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Giambattista Giacomin
- Published
- 2011
46. Critical Point Shift: The Fractional Moment Method
- Author
-
Giambattista Giacomin
- Subjects
Change of measure ,Mathematical analysis ,Random environment ,Granularity ,Rate function ,Critical point (mathematics) ,Mathematics - Abstract
This chapter is devoted to showing that, when α ≥ 1 ∕ 2, quenched and annealed critical points are different for every β > 0, with explicit estimates on the difference. Such a result follows from upper bounds on the free energy that are obtained by estimating fractional moments (of order less than one) of the partition function. Estimates for every β > 0, notably for arbitrarily small values of β, are obtained by using a change of measure argument on the law of the disorder and by coarse graining techniques. Proving such estimates becomes harder and harder as α approaches 1 ∕ 2, i.e. the marginal disorder case in the Harris’ sense: for α = 1 ∕ 2 the Harris criterion yields no prediction and whether quenched and annealed critical points differed or not has been a debated issue in the physical literature.
- Published
- 2011
47. Introduction to Disordered Pinning Models
- Author
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Giambattista Giacomin
- Subjects
Physics ,Delocalized electron ,Partition function (statistical mechanics) ,Statistical mechanics ,Statistical physics ,Concentration inequality ,Energy (signal processing) ,Convexity - Abstract
We introduce the disorder disordered version of the pinning models, both in their quenched and annealed version. We define the free energy of the model and show that also in this case a localization/delocalization transition takes place. Most of the results presented in this chapter may be considered as soft, but they are the result of a subtle, albeit possibly standard in statistical mechanics, way of combining convexity and super-additivity properties. These techniques are repeatedly used in the sequel of these notes.
- Published
- 2011
48. Disorder and Critical Phenomena Through Basic Probability Models
- Author
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École d'été de probabilités de Saint-Flour and Giambattista Giacomin
- Subjects
Basic probability ,Homogeneous ,Critical phenomena ,Granularity ,Statistical physics ,Condensed Matter::Disordered Systems and Neural Networks ,Smoothing ,Critical point (mathematics) ,Mathematics - Abstract
1 Introduction.- 2 Homogeneous pinning systems: a class of exactly solved models.- 3 Introduction to disordered pinning models.- 4 Irrelevant disorder estimates.- 5 Relevant disorder estimates: the smoothing phenomenon.- 6 Critical point shift: the fractional moment method.- 7 The coarse graining procedure.- 8 Path properties.
- Published
- 2011
49. Large scale behavior of semiflexible heteropolymers
- Author
-
Francesco Caravenna, Massimiliano Gubinelli, Giambattista Giacomin, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Institut für Mathematik [Zürich], Universität Zürich [Zürich] = University of Zurich (UZH), Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Université Pierre et Marie Curie - Paris 6 (UPMC), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Caravenna, F, Giacomin, G, Gubinelli, M, Université Paris Dauphine-PSL, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Statistics and Probability ,Scale (ratio) ,82B44 ,Heteropolymer ,Topology ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Probability theory ,Mixing (mathematics) ,Tensor analysi ,60F05 ,FOS: Mathematics ,Disorder ,Statistical physics ,0101 mathematics ,Brownian motion ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,Tensor analysis ,Probability (math.PR) ,010102 general mathematics ,Non-commutative Fourier analysis ,Persistence length ,Large scale limit ,Random walk ,Fick's laws of diffusion ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,60K37 ,43A75 ,Fourier analysis ,symbols ,Semiflexible chain ,Statistics, Probability and Uncertainty ,Tensor calculus ,Mathematics - Probability - Abstract
We consider a general discrete model for heterogeneous semiflexible polymer chains. Both the thermal noise and the inhomogeneous character of the chain (the disorder) are modeled in terms of random rotations. We focus on the quenched regime, i.e., the analysis is performed for a given realization of the disorder. Semiflexible models differ substantially from random walks on short scales, but on large scales a Brownian behavior emerges. By exploiting techniques from tensor analysis and non-commutative Fourier analysis, we establish the Brownian character of the model on large scale and we obtain an expression for the diffusion constant. We moreover give conditions yielding quantitative mixing properties., 24 pages, 1 figure; corrected typos, added French abstract; accepted version to appear in Annales de l'Institut Henri Poincar\'e
- Published
- 2010
50. Dynamical aspects of mean field plane rotators and the Kuramoto model
- Author
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Lorenzo Bertini, Khashayar Pakdaman, Giambattista Giacomin, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), and Benassù, Serena
- Subjects
[MATH.MATH-PR] Mathematics [math]/Probability [math.PR] ,FOS: Physical sciences ,01 natural sciences ,symbols.namesake ,0103 physical sciences ,Statistical physics ,0101 mathematics ,010306 general physics ,ComputingMilieux_MISCELLANEOUS ,Mathematical Physics ,Brownian motion ,Physics ,Operator (physics) ,Kuramoto model ,010102 general mathematics ,Hilbert space ,Time evolution ,Statistical and Nonlinear Physics ,Nonlinear Sciences - Adaptation and Self-Organizing Systems ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Mean field theory ,FOS: Biological sciences ,Quantitative Biology - Neurons and Cognition ,symbols ,Neurons and Cognition (q-bio.NC) ,Spectral gap ,Constant (mathematics) ,Adaptation and Self-Organizing Systems (nlin.AO) - Abstract
The Kuramoto model has been introduced in order to describe synchronization phenomena observed in groups of cells, individuals, circuits, etc... We look at the Kuramoto model with white noise forces: in mathematical terms it is a set of N oscillators, each driven by an independent Brownian motion with a constant drift, that is each oscillator has its own frequency, which, in general, changes from one oscillator to another (these frequencies are usually taken to be random and they may be viewed as a quenched disorder). The interactions between oscillators are of long range type (mean field). We review some results on the Kuramoto model from a statistical mechanics standpoint: we give in particular necessary and sufficient conditions for reversibility and we point out a formal analogy, in the N to infinity limit, with local mean field models with conservative dynamics (an analogy that is exploited to identify in particular a Lyapunov functional in the reversible set-up). We then focus on the reversible Kuramoto model with sinusoidal interactions in the N to infinity limit and analyze the stability of the non-trivial stationary profiles arising when the interaction parameter K is larger than its critical value K_c. We provide an analysis of the linear operator describing the time evolution in a neighborhood of the synchronized profile: we exhibit a Hilbert space in which this operator has a self-adjoint extension and we establish, as our main result, a spectral gap inequality for every K>K_c., 18 pages, 1 figure
- Published
- 2010
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