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We provide a new short and elementary proof of diffusivity for directed polymer in a random environment in the weak disorder phase., Comment: 4 pages

We investigate the upper tail distribution of the partition function of the directed polymer in a random environment on $\mathbb Z^d$ in the weak disorder phase. We show that the distribution of the infinite volume partition function $W^{\beta}_{\infty}$ displays a power-law decay, with an exponent $p^*(\beta)\in [1+\frac{2}{d},\infty)$. We also prove that the distribution of the suprema of the point-to-point and point-to-line partition functions display the same behavior. On the way to these results, we prove a technical estimate of independent interest: the $L^p$-norm of the partition function at the time when it overshoots a high value $A$ is comparable to $A$. We use this estimate to extend the validity of many recent results that were proved under the assumption that the environment is upper bounded.

We investigate a quadratic dynamical system known as nonlinear recombinations. This system models the evolution of a probability measure over the Boolean cube, converging to the stationary state obtained as the product of the initial marginals. Our main result reveals a cutoff phenomenon for the total variation distance in both discrete and continuous time. Additionally, we derive the explicit cutoff profiles in the case of monochromatic initial distributions. These profiles are different in the discrete and continuous time settings. The proof leverages a pathwise representation of the solution in terms of a fragmentation process associated to a binary tree. In continuous time, the underlying binary tree is given by a branching random process, thus requiring a more elaborate probabilistic analysis., Comment: 35 pages, 4 figures. Minor corrections, some references added

We show that if the normalized partition function $W^{\beta}_n$ of the directed polymer model on $\mathbb{Z}^d$ converges to zero, then it does so exponentially fast. This implies that there exists a critical temperature $\beta_c$ such that the renormalized partition function has a non-degenerate limit for all $\beta\in [0,\beta_c]$ -- weak disorder holds -- while for $\beta\in (\beta_c,\infty)$ it converges exponentially fast to zero -- very strong disorder holds. This solves a twenty-years-old conjecture formulated by Comets, Yoshida, Carmona and Hu. Our proof requires a technical assumption on the environment, namely, that it is bounded from above., Comment: 28 pages. Fixes a couple of typos present in v1

Given $d\ge 1$, we provide a construction of the random measure - the critical Gaussian Multiplicative Chaos - formally defined $e^{\sqrt{2d}X}\mathrm{d} \mu$ where $X$ is a $\log$-correlated Gaussian field and $\mu$ is a locally finite measure on $\mathbb R^d$. Our construction generalizes the one performed in the case where $\mu$ is the Lebesgue measure. It requires that the measure $\mu$ is sufficiently spread out, namely that for $\mu$ almost every $x$ we have $$ \int_{B(0,1)}\frac{\mu(\mathrm{d} y)}{|x-y|^{d}e^{\rho\left(\log \frac{1}{|x-y|} \right)}}<\infty, $$ for any compact set where $\rho:\mathbb R_+\to \mathbb R_+$ can be chosen to be any lower envelope function for the $3$-Bessel process (this includes $\rho(x)=x^{\alpha}$ with $\alpha\in (0,1/2)$). We prove that three distinct random objects converge to a common limit which defines the critical GMC: the derivative martingale, the critical martingale, and the exponential of the mollified field. We also show that the above criterion for the measure $\mu$ is in a sense optimal., Comment: 32 pages

The complex Gaussian Multiplicative Chaos (or complex GMC) is informally defined as a random measure $e^{\gamma X} \mathrm{d} x$ where $X$ is a log correlated Gaussian field on $\mathbb R^d$ and $\gamma=\alpha+i\beta$ is a complex parameter. The correlation function of $X$ is of the form $$ K(x,y)= \log \frac{1}{|x-y|}+ L(x,y),$$ where $L$ is a continuous function. In the present paper, we consider the cases $\gamma\in \mathcal P_{\mathrm{I/II}}$ and $\gamma\in \mathcal{P}'_{\mathrm{II/III}}$ where $$ \mathcal P_{\mathrm{I/II}}:= \{ \alpha+i \beta \ : \alpha,\beta \in \mathbb R \ ; |\alpha|>|\beta| \ ; \ |\alpha|+|\beta|=\sqrt{2d} \}, $$ and $$ \mathcal{P}'_{\mathrm{II/III}}:= \{ \alpha+i \beta \ : \alpha,\beta \in \mathbb R \ ; \ |\alpha|= \sqrt{d/2} \ ; \ |\beta|>\sqrt{2d} \},$$ We prove that if $X$ is replaced by an approximation $X_\epsilon$ obtained via mollification, then $e^{\gamma X_\epsilon} \mathrm{d} x$, when properly rescaled, converges when $\epsilon\to 0$. The limit does not depend on the mollification kernel. When $\gamma\in \mathcal P_{\mathrm{I/II}}$, the convergence holds in probability and in $L^p$ for some value of $p\in [1,\sqrt{2d}/\alpha)$. When $\gamma\in \mathcal{P}'_{\mathrm{II/III}}$ the convergence holds only in law. In this latter case, the limit can be described a complex Gaussian white noise with a random intensity given by a critical real GMC. The regions $\mathcal P_{\mathrm{I/II}}$ and $ \mathcal{P}'_{\mathrm{II/III}}$ correspond to phase boundary between the three different regions of the complex GMC phase diagram. These results complete previous results obtained for the GMC in phase I and III and only leave as an open problem the question of convergence in phase II., Comment: 50 pages, 1 figure

We present new, short and self-contained proofs of the convergence (with an adequate renormalization) of four different sequences to the critical Gaussian Multiplicative Chaos:(a) the derivative martingale (b) the critical martingale (c) the exponential of the mollified field (d) the subcritical Gaussian Multiplicative Chaos., Comment: 26 pages, 5 figures

We study the stochastic heat equation (SHE) $\partial_t u = \frac12 \Delta u + \beta u \xi$ driven by a multiplicative L\'evy noise $\xi$ with positive jumps and amplitude $\beta>0$, in arbitrary dimension $d\geq 1$. We prove the existence of solutions under an optimal condition if $d=1,2$ and a close-to-optimal condition if $d\geq3$. Under an assumption that is general enough to include stable noises, we further prove that the solution is unique. By establishing tight moment bounds on the multiple L\'evy integrals arising in the chaos decomposition of $u$, we further show that the solution has finite $p$th moments for $p>0$ whenever the noise does. Finally, for any $p>0$, we derive upper and lower bounds on the moment Lyapunov exponents of order $p$ of the solution, which are asymptotically sharp in the limit as $\beta\to0$. One of our most striking findings is that the solution to the SHE exhibits a property called strong intermittency (which implies moment intermittency of all orders $p>1$ and pathwise mass concentration of the solution), for any non-trivial L\'evy measure, at any disorder intensity $\beta>0$, in any dimension $d\geq1$.

We survey recent results concerning the total-variation mixing time of the simple exclusion process on the segment (symmetric and asymmetric) and a continuum analog, the simple random walk on the simplex with an emphasis on cutoff results. A Markov chain is said to exhibit cutoff if on a certain time scale, the distance to equilibrium drops abruptly from $1$ to $0$. We also review a couple of techniques used to obtain these results by exposing and commenting some elements of proof., Comment: Short survey paper. 24 pages, 3 figures

We consider the simple exclusion process in the integer segment $ [1, N]$ with $k\le N/2$ particles and spatially inhomogenous jumping rates. A particle at site $x\in [ 1, N]$ jumps to site $x-1$ (if $x\ge 2$) at rate $1-\omega_x$ and to site $x+1$ (if $x \le N-1$) at rate $\omega_x$ if the target site is not occupied. The sequence $\omega=(\omega_x)_{ x \in \mathbb{Z}}$ is chosen by IID sampling from a probability law whose support is bounded away from zero and one (in other words the random environment satisfies the uniform ellipticity condition). We further assume $\mathbb{E}[ \log \rho_1 ]<0$ where $\rho_1:= (1-\omega_1)/\omega_1$, which implies that our particles have a tendency to move to the right. We prove that the mixing time of the exclusion process in this setup grows like a power of $N$. More precisely, for the exclusion process with $N^{\beta+o(1)}$ particles where $\beta\in [0,1)$, we have in the large $N$ asymptotic $$ N^{\max\left(1,\frac {1}{\lambda}, \beta+ \frac 1 {2\lambda}\right)+o(1)} \le t_{\mathrm{Mix}}^{N,k} \le N^{C+o(1)}$$ where $\lambda>0$ is such that $\mathbb{E}[\rho_1^{\lambda}]=1$ ($\lambda=\infty$ if the equation has no positive root) and $C$ is a constant which depends on the distribution of $\omega$. We conjecture that our lower bound is sharp up to sub-polynomial correction., Comment: 35 pages, 5 figures

Gaussian Multiplicative Chaos (GMC) is informally defined as a random measure $e^{\gamma X} \mathrm{d} x$ where $X$ is Gaussian field on $\mathbb R^d$ (or an open subset of it) whose correlation function is of the form $ K(x,y)= \log \frac{1}{|y-x|}+ L(x,y),$ where $L$ is a continuous function $x$ and $y$ and $\gamma=\alpha+i\beta$ is a complex parameter. In the present paper, we consider the case $\gamma\in \mathcal P'_{\mathrm{III}}$ where $$ \mathcal P'_{\mathrm{III}}:= \{ \alpha+i \beta \ : \alpha,\gamma \in \mathbb R , \ |\alpha|<\sqrt{d/2}, \ \alpha^2+\beta^2\ge d \}.$$ We prove that if $X$ is replaced by the approximation $X_\varepsilon$ obtained by convolution with a smooth kernel, then $e^{\gamma X_\varepsilon} \mathrm d x$, when properly rescaled, has an explicit non-trivial limit in distribution when $\varepsilon$ goes to zero. This limit does not depend on the specific convolution kernel which is used to define $X_{\varepsilon}$ and can be described as a complex Gaussian white noise with a random intensity given by a real GMC associated with parameter $2\alpha$., Comment: 35 pages. Stable convergence and references added

In this paper, we study the so-called intermediate disorder regime for a directed polymer in a random environment with heavy-tail. Consider a simple symmetric random walk $(S_n)_{n\geq 0}$ on $\mathbb{Z}^d$, with $d\geq 1$, and modify its law using Gibbs weights in the product form $\prod_{n=1}^{N} (1+\beta\eta_{n,S_n})$, where $(\eta_{n,x})_{n\ge 0, x\in \mathbb{Z}^d}$ is a field of i.i.d. random variables whose distribution satisfies $\mathbb{P}(\eta>z) \sim z^{-\alpha}$ as $z\to\infty$, for some $\alpha\in(0,2)$. We prove that if $\alpha< \min(1+\frac{2}{d},2)$, when sending $N$ to infinity and rescaling the disorder intensity by taking $\beta=\beta_N \sim N^{-\gamma}$ with $\gamma =\frac{d}{2\alpha}(1+\frac{2}{d}-\alpha)$, the distribution of the trajectory under diffusive scaling converges in law towards a random limit, which is the continuum polymer with L\'evy $\alpha$-stable noise constructed in the companion paper arXiv:2007.06484., Comment: 48 pages, comments are welcome

We consider the Gibbs sampler, or heat bath dynamics associated to log-concave measures on $\mathbb{R}^N$ describing $\nabla\varphi$ interfaces with convex potentials. Under minimal assumptions on the potential, we find that the spectral gap of the process is always given by $\mathrm{gap}_N=1-\cos(\pi/N)$, and that for all $\epsilon\in(0,1)$, its $\epsilon$-mixing time satisfies $T_N(\epsilon)\sim \frac{\log N}{2\mathrm{gap}_N}$ as $N\to\infty$, thus establishing the cutoff phenomenon. The results reveal a universal behavior in that they do not depend on the choice of the potential., Comment: 38 pages

We study the dynamical behavior of a one dimensional interface interacting with a sticky unpenetrable substrate or wall. The interface is subject to two effects going in opposite directions. Contact between the interface and the substrate are given an energetic bonus while an external force with constant intensity pulls the interface away from the wall. Our interface is modeled by the graph of a one-dimensional nearest-neighbor path on $\mathbb{Z}_+$, starting at $0$ and ending at $0$ after $2N$ steps, the wall corresponding to level-zero the horizontal axis. At equilibrium each path $\xi=(\xi_x)_{x=0}^{2N}$, is given a probability proportional to $\lambda^{H(\xi)} \exp(\frac{\sigma}{N}A(\xi))$, where $H(\xi):=\#\{x \ : \xi_x=0\}$ and $A(\xi)$ is the area enclosed between the path $\xi$ and the $x$-axis. We then consider the classical heat-bath dynamics which equilibrates the value of each $\xi_x$ at a constant rate via corner-flip. Investigating the statics of the model, we derive the full phase diagram in $\lambda$ and $\sigma$ of this model, and identify the critical line which separates a localized phase where the pinning force sticks the interface to the wall and a delocalized one, for which the external force stabilizes $\xi$ around a deterministic shape at a macroscopic distance of the wall. On the dynamical side, we identify a second critical line, which separates a rapidly mixing phase (for which the system mixes in polynomial time) to a slow phase where the mixing time grows exponentially. In this slowly mixing regime we obtain a sharp estimate of the mixing time on the $\log$ scale, and provide evidences of a metastable behavior., Comment: 40 pages, 6 Figures

We present in this paper the construction of a continuum directed polymer model in an environment given by space-time L\'evy noise. One of the main objectives of this construction is to describe the scaling limit of discrete directed polymer in an heavy-tail environment and for this reason we put special emphasis on the case of $\alpha$-stable noises with $\alpha \in (1,2)$. Our construction can be performed in arbitrary dimension, provided that the L\'evy measure satisfies specific (and dimension dependent) conditions. We also discuss a few basic properties of the continuum polymer and the relation between this model and the Stochastic Heat Equation with multiplicative L\'evy noise., Comment: 60 pages, some typos corrected. In the last version, an appendix on the convergence of truncated L\'evy noises has been added, together with other various comments

In the present paper, we show that (under some minor technical assumption) Complex Gaussian Multiplicative Chaos defined as the complex exponential of a $\log$-correlated Gaussian field can be obtained by taking the limit of the exponential of the field convoluted with a smoothing Kernel. We consider two types of chaos: $e^{\gamma X}$ for a log correlated field $X$ and $\gamma=\alpha+i\beta$, $\alpha, \beta\in \mathbb R$ and $e^{\alpha X+i\beta Y}$ for $X$ and $Y$ two independent fields with $\alpha, \beta\in \mathbb R$. Our result is valid in the range $$ \mathcal O_{\mathrm{sub}}:=\{ \alpha^2+\beta^2

We investigate the localization transition for a simple model of interface which interacts with an inhomonegeous defect plane. The interface is modeled by the graph of a function $\phi: \mathbb Z^2 \to \mathbb Z$,and the disorder is given by a fixed realization of a field of IID centered random variables$(\omega_x)_{x\in \mathbb Z^2}$. The Hamiltonian of the system depends on three parameters $\alpha,\beta>0$ and $h\in \mathbb R$ which determine respectively the intensity of nearest neighbor interaction the amplitude of disorder and the mean value of the interaction with the substrate, and is given by the expression $$\mathcal H(\phi):= \beta\sum_{x\sim y} |\phi(x)-\phi(y)|- \sum_{x} (\alpha\omega_x+h){\bf 1}_{\{\phi(x)=0\}}.$$ We focus on the large-$\beta$/rigid phase phase of the Solid-On-Solid (SOS) model. In that regime, we provide a sharp description of the phase transition in $h$ from a localized phase to a delocalized one corresponding respectivelly to a positive and vanishing fraction of points with $\phi(x)=0$. We prove that the critical value for $h$ corresponds to that of the annealed model and is given by $h_c(\alpha)= -\log \mathbb E[e^{\alpha \omega}]$, and that near the critical point, the free energy displays the following critical behavior $$F_\beta(\alpha,h_c+u )\stackrel{u\to 0+}{\sim} \max_{n\ge 1} \left\{\theta_1 e^{-4\beta n} u- \frac{1}{2}\theta^2_1 e^{-8\beta n} \frac{\mathrm{Var}\left[e^{\alpha \omega}\right]}{\mathbb E \left[ e^{\alpha \omega} \right]^2}\right\}.$$ The positive constant $\theta_1(\beta)>0$ is defined by the asymptotic probability of spikes for the infinite volume SOS with $0$ boundary condition $\theta_1(\beta):=\lim_{n\to \infty} e^{4\beta n}\mathbf P_{\beta} (\phi({\bf 0})=n)$ ..., Comment: 44 pages 6 figures (revised version)

We continue the study, initiated in [Giacomin and Lacoin, JEMS 2018], of the localization transition of a lattice free field $\phi=(\phi(x))_{x \in Z^d}$, $d\ge 3$, in presence of a quenched disordered substrate. The presence of the substrate affects the interface at the spatial sites in which the interface height is close to zero. This corresponds to the Hamiltonian $$ \sum_{x\in Z^d }(\beta \omega_x+h)\delta_x,$$ where $\delta_x=1_{[-1,1]}(\phi(x))$, and $(\omega_x)_{x\in Z^d}$ is an IID centered field. A transition takes place when the average pinning potential $h$ goes past a threshold $h_c(\beta)$: from a delocalized phase $hh_c(\beta)$ where the field sticks to the substrate. In [Giacomin and Lacoin, JEMS 2018] the critical value of $h$ is identified and it coincides, up to the sign, with the $\log$-Laplace transform of $\omega=\omega_x$, that is $-h_c(\beta)=\lambda(\beta):=\log E[e^{\beta\omega}]$. Here we obtain the sharp critical behavior of the free energy approaching criticality: $$\lim_{u\searrow 0} \frac{ F(\beta,h_c(\beta)+u)}{u^2}= \frac{1}{2\, \textrm{Var}\left(e^{\beta \omega-\lambda(\beta)}\right)}.$$ Moreover, we give a precise description of the trajectories of the field in the same regime: the absolute value of the field is $\sqrt{2\sigma_d^2\vert\log(h-h_c(\beta))\vert}$ to leading order when $h\searrow h_c(\beta)$ except on a vanishing fraction of sites ($\sigma_d^2$ is the single site variance of the free field)., Comment: 62 pages

By viewing the $N$-simplex as the set of positions of $N-1$ ordered particles on the unit interval, the adjacent walk is the continuous time Markov chain obtained by updating independently at rate 1 the position of each particle with a sample from the uniform distribution over the interval given by the two particles adjacent to it. We determine its spectral gap and prove that both the total variation distance and the separation distance to the uniform distribution exhibit a cutoff phenomenon, with mixing times that differ by a factor $2$. The results are extended to the family of log-concave distributions obtained by replacing the uniform sampling by a symmetric log-concave Beta distribution.

A rigorous probabilistic construction of Liouville conformal field theory (LCFT) on the Riemann sphere was recently given by David-Kupiainen and the last two authors. In this paper, we focus on the connection between LCFT and the classical Liouville field theory via the semiclassical approach. LCFT depends on a parameter $\gamma \in (0,2)$ and the limit $\gamma \to 0$ corresponds to the semiclassical limit of the theory. Within this asymptotic and under a negative curvature condition (on the limiting metric of the theory), we determine the limit of the correlation functions and of the associated Liouville field. We also establish a large deviation result for the Liouville field: as expected, the large deviation functional is the classical Liouville action. As a corollary, we give a new (probabilistic) proof of the Takhtajan-Zograf theorem which relates the classical Liouville action (taken at its minimum) to Poincar\'e's accessory parameters. Finally, we gather conjectures in the positive curvature case (including the study of the so-called quantum spheres introduced by Duplantier-Miller-Sheffield).

The Sine-Gordon model is obtained by tilting the law of a log-correlated Gaussian field $X$ defined on a subset of $\mathbb{R}^d$ by the exponential of its cosine, namely $\exp(\alpha \smallint \cos (\beta X))$. It is an important model in quantum field theory or in statistic physics like in the study of log-gases. In spite of its relatively simple definition, the model has a very rich phenomenology. While the integral $\smallint \cos (\beta X)$ can properly be defined when $\beta^2

It is a fact simple to establish that the mixing time of the simple random walk on a d-regular graph $G_n$ with n vertices is asymptotically bounded from below by $d/ ((d-2)\log (d-1))\log n$. Such a bound is obtained by comparing the walk on $G_n$ to the walk on the infinite $d$-regular tree. If one can map another infinite transitive graph onto $G_n$, then we can improve the strategy by using a comparison with the random walk on this transitive graph (instead of that of the regular tree), and we obtain a lower bound of the form $1/h \log n$, where $h$ is the entropy rate associated with the walk on the transitive graph. We call this the entropic lower bound. It was recently proved that in the case of the tree, this entropic lower bound is sharp when graphs have minimal spectral radius and thus that in that case the random walk exhibit cutoff at the entropic time. In this paper, we provide a generalization of the result by providing a sufficient condition on the spectra the random walks on $G_n$ under which the random walk exhibit cutoff at the entropic time. It applies notably to anisotropic random walks on random $d$-regular graphs and to random walks on random $n$-lifts of a base graph (including non-reversible walks)., Comment: 45 pages, non-reversible random walks added, accepted for publication in Journal of the Institute of Mathematics of Jussieu

We consider the Solid-On-Solid model interacting with a wall, which is the statistical mechanics model associated with the integer-valued field $(\phi(x))_{x\in \mathbb Z^2}$, and the energy functional $$V(\phi)=\beta \sum_{x\sim y}|\phi(x)-\phi(y)|-\sum_{x}\left( h{\bf 1}_{\{\phi(x)=0\}}-\infty{\bf 1}_{\{\phi(x)<0\}} \right).$$ We prove that for $\beta$ sufficiently large, there exists a decreasing sequence $(h^*_n(\beta))_{n\ge 0}$, satisfying $\lim_{n\to\infty}h^*_n(\beta)=h_w(\beta),$ and such that: $(A)$ The free energy associated with the system is infinitely differentiable on $\mathbb R \setminus \left(\{h^*_n\}_{n\ge 1}\cup h_w(\beta)\right)$, and not differentiable on $\{h^*_n\}_{n\ge 1}$. $(B)$ For each $n\ge 0$ within the interval $(h^*_{n+1},h^*_n)$ (with the convention $h^*_0=\infty$), there exists a unique translation invariant Gibbs state which is localized around height $n$, while at a point of non-differentiability, at least two ergodic Gibbs state coexist. The respective typical heights of these two Gibbs states are $n-1$ and $n$. The value $h^*_n$ corresponds thus to a first order layering transition from level $n$ to level $n-1$. These results combined with those obtained in [23] provide a complete description of the wetting and layering transition for SOS., Comment: 56 pages, 3 Figures, references added, minor error corrected

The generalized 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 $1+ \alpha\geq 1$. 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 \emph{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., Comment: 26 pages, 1 figure

We provide a complete description of the low temperature wetting transition for the two dimensional Solid-On-Solid model. More precisely we study the integer-valued field $(\phi(x))_{x\in \mathbb Z^2}$, associated associated to the energy functional $$V(\phi)=\beta \sum_{x\sim y}|\phi(x)-\phi(y)|-\sum_{x}\left(h{\bf 1}_{\{\phi(x)=0\}}-\infty{\bf 1}_{\{\phi(x)<0\}} \right).$$ It is known since the pioneering work of Chalker (J. Phys. A {\bf 15} (1982) 481-485) that for every $\beta$, there exists $h_{w}(\beta)>0$ delimiting a transition between a delocalized phase ($hh_{w}(\beta)$) where this proportion is positive. We prove in the present paper that for $\beta$ sufficiently large we have $$h_w(\beta)= \log \left(\frac{e^{4\beta}}{e^{4\beta}-1}\right).$$ Furthermore we provide a sharp asymptotic for the free energy at the vicinity of the critical point: We show that close to $h_w(\beta)$, the free energy is approximately piecewise affine and that the points of discontinuity for the derivative of the affine approximation forms a geometric sequence accumulating on the right of $h_w(\beta)$. This asymptotic behavior provides a strong evidence for the conjectured existence of countably many "layering transitions" at the vicinity of the critical point, corresponding to jumps for the typical height of the field., Comment: 39 pages, 4 Figures

We investigate disorder relevance for the pinning of a renewal when the law of the random environment is in the domain of attraction of a stable law with parameter $\gamma \in (1,2)$. Assuming that the renewal jumps have power-law decay, we determine under which condition the critical point of the system modified by the introduction of a small quantity of disorder. In an earlier study of the problem, we have shown that the answer depends on the value of the tail exponent $\alpha$ associated to the distribution of renewal jumps: when $\alpha>1-\gamma^{-1}$ a small amount of disorder shifts the critical point whereas it does not when $\alpha<1-\gamma^{-1}$. The present paper is focused on the boundary case $\alpha=1-\gamma^{-1}$. We show that a critical point shifts occurs in this case, and obtain an estimate for its intensity., Comment: 17 pages

We consider the biased card shuffling and the Asymmetric Simple Exclusion Process (ASEP) on the segment. We obtain the asymptotic of their mixing times: our result show that these two continuous-time Markov chains display cutoff. Our analysis combines several ingredients including: a study of the hydrodynamic profile for ASEP, the use of monotonic eigenfunctions, stochastic comparisons and concentration inequalities., Comment: 50 pages

We investigate disorder relevance for the pinning of a renewal whose inter-arrival law has tail exponent $\alpha>0$ when the law of the random environment is in the domain of attraction of a stable law with parameter $\gamma \in (1,2)$. We prove that in this case, the effect of disorder is not decided by the sign of the specific heat exponent as predicted by Harris criterion but that a new criterion emerges to decide disorder relevance. More precisely we show that when $\alpha>1-\gamma^{-1}$ there is a shift of the critical point at every temperature whereas when $\alpha< 1-\gamma^{-1}$, at high temperature the quenched and annealed critical point coincide, and the critical exponents are identical., Comment: 25 Pages

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

This paper continues a study initiated in [34], on the localization transition of a lattice free field on $\mathbb Z^d$ interacting with a quenched disordered substrate that acts on the interface when its height is close to zero. The substrate has the tendency to localize or repel the interface at different sites. A transition takes place when the average pinning potential $h$ goes past a threshold $h_c$: from a delocalized phase $hh_c$ where the field sticks to the substrate. Our goal is to investigate the effect of the presence of disorder on this phase transition. We focus on the two dimensional case $(d=2)$ for which we had obtained so far only limited results. We prove that the value of $h_c(\beta)$ is the same as for the annealed model, for all values of $\beta$ and that in a neighborhood of $h_c$. Moreover we prove that in contrast with the case $d\ge 3$ where the free energy has a quadratic behavior near the critical point, the phase transition is of infinite order $$\lim_{u\to 0+} \frac{ \log \mathrm{F}(\beta,h_c(\beta)+u)}{(\log u)}= \infty.$$, Comment: 60 pages, 1 Figure, minor corrections

The cutoff phenomenon describes the case when an abrupt transition occurs in the convergence of a Markov chain to its equilibrium measure. There are various metrics which can be used to measure the distance to equilibrium, each of which corresponding to a different notion of cutoff. The most commonly used are the total-variation and the separation distances. In this note we prove that the cutoff for these two distances are not equivalent by constructing several counterexamples which display cutoff in total-variation but not in separation and with the opposite behavior, including lazy simple random walk on a sequence of uniformly bounded degree expander graphs. These examples give a negative answer to a question of Ding, Lubetzky and Peres., Comment: 37 pages, 9 figures. Details added for some proofs and minor corrections

We investigate the high-temperature behavior of the directed polymer model in dimension $1+2$. More precisely we study the difference $\Delta \mathtt{F}(\beta)$ between the quenched and annealed free energies for small values of the inverse temperature $\beta$. This quantity is associated to localization properties of the polymer trajectories, and is related to the overlap fraction of two replicas. Adapting recent techniques developed by the authors in the context of the disordered pinning model (Berger and Lacoin, arXiv:1503.07315 [math-ph]), we identify the sharp asymptotic high temperature behavior \[\lim_{\beta\to 0} \, \beta^2 \log \Delta \mathtt{F}(\beta) = -\pi \, .\], Comment: 19 pages

The effect of disorder for pinning models is a subject which has attracted much attention in theoretical physics and rigorous mathematical physics. A peculiar point of interest is the question of coincidence of the quenched and annealed critical point for a small amount of disorder. The question has been mathematically settled in most cases in the last few years, giving in particular a rigorous validation of the Harris Criterion on disorder relevance. However, the marginal case, where the return probability exponent is equal to $1/2$, i.e. where the inter-arrival law of the renewal process is given by $K(n)=n^{-3/2}\phi(n)$ where $\phi$ is a slowly varying function, has been left partially open. In this paper, we give a complete answer to the question by proving a simple necessary and sufficient criterion on the return probability for disorder relevance, which confirms earlier predictions from the literature. Moreover, we also provide sharp asymptotics on the critical point shift: in the case of the pinning (or wetting) of a one dimensional simple random walk, the shift of the critical point satisfies the following high temperature asymptotics $$ \lim_{\beta\rightarrow 0}\beta^2\log h_c(\beta)= - \frac{\pi}{2}. $$ This gives a rigorous proof to a claim of B. Derrida, V. Hakim and J. Vannimenus (Journal of Statistical Physics, 1992)., Comment: 34 Pages

In this paper, we give a very accurate description of the way the simple exclusion process relaxes to equilibrium. Let $P_t$ denote the semi-group associated the exclusion on the circle with $2N$ sites and $N$ particles. For any initial condition $\chi$, and for any $t\ge\frac{4N^2}{9\pi ^2}\log N$, we show that the probability density $P_t(\chi,\cdot)$ is given by an exponential tilt of the equilibrium measure by the main eigenfunction of the particle system. As $\frac{4N^2}{9\pi^2}\log N$ is smaller than the mixing time which is $\frac{N^2}{2\pi^2}\log N$, this allows to give a sharp description of the cutoff profile: if $d_N(t)$ denote the total-variation distance starting from the worse initial condition we have \[\lim_{N\to\infty}d_N\biggl(\frac{N^2}{2\pi^2}\log N+\frac{N^2}{\pi^2}s\biggr)=\operatorname {erf}\biggl(\frac{\sqrt{2}}{\pi}e^{-s}\biggr),\] where $\operatorname {erf}$ is the Gauss error function., Comment: Published at http://dx.doi.org/10.1214/15-AOP1053 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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., Comment: 55 pages, 1 figure. Added a statement on path properties, corrected misprints and reformulated some arguments

In this note, we construct an example of a sequence of $n$-fold product chains which does not display cutoff for total-variation distance neither for separation distance. In addition we show that this type of product chains necessarily displays pre-cutoff., Comment: 9 Pages, 1 Figure

The presence of frozen-in or quenched disorder in a system can often modify the nature of its phase transition. A particular instance of this phenomenon is the so-called rounding effect: it has been shown in many cases that the free-energy curve of the disordered system at its critical point is smoother than that of the homogenous one. In particular some disordered systems do not allow first-order transitions. We study this phenomenon for the pinning of a renewal with stretched-exponential tails on a defect line (the distribution $K$ of the renewal increments satisfies $K(n) \sim c_K\exp(-n^{\alpha}),$ $\alpha\in (0,1)$) which has a first order transition when disorder is not present. We show that the critical behavior of the disordered system depends on the value of $\alpha$: when $\alpha>1/2$ the transition remains first order, whereas the free-energy diagram is smoothed for $\alpha\le 1/2$. Furthermore we show that the rounding effect is getting stronger when $\alpha$ diminishes., Comment: 20 pages, 2 Figure, a few minor errors corrected

In this paper, we investigate the mixing time of the simple exclusion process on the circle with $N$ sites, with a number of particle $k(N)$ tending to infinity, both from the worst initial condition and from a typical initial condition. We show that the worst-case mixing time is asymptotically equivalent to $(8\pi^2)^{-1}N^2\log k$, while the cutoff window, is identified to be $N^2$. Starting from a typical condition, we show that there is no cutoff and that the mixing time is of order $N^2$., Comment: 37 pages, 3 Figures

Liouville Field Theory (LFT for short) is a two dimensional model of random surfaces, which is for instance involved in $2d$ string theory or in the description of the fluctuations of metrics in $2d$ Liouville quantum gravity. This is a probabilistic model that consists in weighting the classical Free Field action with an interaction term given by the exponential of a Gaussian multiplicative chaos. The main input of our work is the study of the semiclassical limit of the theory, which is a prescribed asymptotic regime of LFT of interest in physics literature (see \cite{witten} and references therein). We derive exact formulas for the Laplace transform of the Liouville field in the case of flat metric on the unit disk with Dirichlet boundary conditions. As a consequence, we prove that the Liouville field concentrates on the solution of the classical Liouville equation with explicit negative scalar curvature. We also characterize the leading fluctuations, which are Gaussian and massive, and establish a large deviation principle. Though considered as an ansatz in the whole physics literature, it seems that it is the first rigorous probabilistic derivation of the semiclassical limit of LFT. On the other hand, we carry out the same analysis when we further weight the Liouville action with heavy matter operators. This procedure appears when computing the $n$-points correlation functions of LFT., Comment: 42 pages; 3 figures; Typos corrected

In this paper we investigate the dynamical behavior of an interface or polymer, in interaction with a distant attractive substrate. The interface is modeled by the graph of a nearest neighbor path with non-negative integer coordinates, and the equilibrium measure associates to each path \eta\ a probability proportional to \lambda^{H(\eta)} where \lambda\ is non-negative and H(\eta) is the number of contacts between \eta\ and the substrate at zero. The dynamics is the natural "spin flip" dynamics associated to this equilibrium measure. We let the distance to the substrate at both polymer ends be equal to aN, where 0 < a < 1/2 is a fixed parameter, and N is the length the system. With this setup, we show that the dynamical behavior of the system crucially depends on \lambda: when \lambda\ \leq 2/(1-2a) we show that the system only needs a time which is polynomial in N to reach its equilibrium state, whereas if \lambda\ > 2/(1-2a) the mixing time is exponential in N and the system relaxes in an exponential manner which is typical of metastability., Comment: 23 pages, 5 figures