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The current standard model of cosmology successfully describes a variety of measurements, but the nature of its main ingredients, dark matter and dark energy, remains unknown. Euclid is a medium-class mission in the Cosmic Vision 2015-2025 programme of the European Space Agency (ESA) that will provide high-resolution optical imaging, as well as near-infrared imaging and spectroscopy, over about 14,000 deg^2 of extragalactic sky. In addition to accurate weak lensing and clustering measurements that probe structure formation over half of the age of the Universe, its primary probes for cosmology, these exquisite data will enable a wide range of science. This paper provides a high-level overview of the mission, summarising the survey characteristics, the various data-processing steps, and data products. We also highlight the main science objectives and expected performance., Comment: Accepted for publication in the A&A special issue`Euclid on Sky'

Let $N$ be a norm on $\mathbb{R}^d$ with $d\geq 2$ and consider $\chi$ a homogeneous Poisson point process on $\mathbb{R}^d$ with intensity $\varepsilon\in [0,\infty)$. We define the Boolean model $\Sigma_{N, \varepsilon}$ as the union of the balls of diameter $1$ for the norm $N$ and centered at the points of $\chi$. For every $x,y \in \mathbb{R}^d$, Let $T_{N, \varepsilon} (x,y)$ be the minimum time needed to travel from $x$ to $y$ if one travels at speed $1$ outside $\Sigma_{N,\varepsilon}$ and at infinite speed inside $\Sigma_{N,\varepsilon}$: this defines a continuous model of first-passage percolation, that has been studied in \cite{GT17,GT22} for $N=\| \cdot \|_2$, the Euclidean norm. The exact calculation of the time constant of this model $\mu_{N,\varepsilon} (x):=\lim_{n\rightarrow \infty}T_{N,\varepsilon} (0,n x) / n $ is out of reach. We investigate here the behavior of $\varepsilon \mapsto \mu_{N,\varepsilon} (x)$ near $0$, and enlight how the speed at which $N(x) - \mu_{N,\varepsilon} (x) $ goes to $0$ depends on $x$ and $N$. For instance, if $N$ is the $p$-norm for $p\in (1,\infty)$, we prove that $N(x) - \mu_{\| \cdot \|_p,\epsilon} (x)$ is of order $\varepsilon ^{\kappa_p(x)}$ with $$\kappa_p(x): = \frac{1}{d- \frac{d_1(x)-1}{2} - \frac{d-d_1 (x)}{p}}\,,$$where $d_1(x)$ is the number of non null coordinates of $x$.Together with the study of the time constant, we also prove a control on the $N$-length of the geodesics, and get some informations on the number of points of $\chi$ really useful to those geodesics. The results are in fact more natural to prove in a slightly different setting, where instead of centering a ball of diameter $1$ at each point of $\chi$ and traveling at infinite speed inside $\Sigma_{N\varepsilon}$, we put a reward of one unit of time at those points.

We study the concepts of complete positivity, positivity and non-Markovianity in a two-level open quantum system whose dynamics are governed by a time-local quantum master equation. We establish necessary and sufficient conditions on the time-dependent relaxation rates to ensure complete positivity and positivity of the dynamical map. We discuss their relations with the non-Markovian behavior of the open system. We also analyze the long-time asymptotic behavior of the dynamics as a function of the rates. We show under which conditions on the rates the system tends to the equilibrium state. Different examples illustrate this general study., Comment: 22 pages, 2 figures

We consider the standard model of first-passage percolation on $\mathbb{Z}^d$ ($d\geq 2$), with i.i.d. passage times associated with either the edges or the vertices of the graph. We focus on the particular case where the distribution of the passage times is the Bernoulli distribution with parameter $1-\epsilon$. These passage times induce a random pseudo-metric $T_\epsilon$ on $\mathbb{R}^d$. By subadditive arguments, it is well known that for any $z\in\mathbb{R}^d\setminus \{0\}$, the sequence $T_\epsilon (0,\lfloor nz \rfloor) / n$ converges a.s. towards a constant $\mu_\epsilon (z)$ called the time constant. We investigate the behavior of $\epsilon \mapsto \mu_\epsilon (z)$ near $0$, and prove that $\mu_\epsilon (z) = \| z\|_1 - C (z) \epsilon ^{1/d_1(z)} + o ( \epsilon ^{1/d_1(z)}) $, where $d_1(z)$ is the number of non null coordinates of $z$, and $C(z)$ is a constant whose dependence on $z$ is partially explicit.

We consider the standard first passage percolation model in the rescaled lattice $\mathbb Z^d/n$ for $d\geq 2$ and a bounded domain $\Omega$ in $\mathbb R^d$. We denote by $\Gamma^1$ and $\Gamma^2$ two disjoint subsets of $\partial \Omega$ representing respectively the sources and the sinks, \textit{i.e.}, where the water can enter in $\Omega$ and escape from $\Omega$. A cutset is a set of edges that separates $\Gamma ^1$ from $\Gamma^2$ in $\Omega$, it has a capacity given by the sum of the capacities of its edges. Under some assumptions on $\Omega$ and the distribution of the capacities of the edges, we already know a law of large numbers for the sequence of minimal cutsets $(\mathcal E_n^{min})_{n\geq 1}$: the sequence $(\mathcal E_n^{min})_{n\geq 1}$ converges almost surely to the set of solutions of a continuous deterministic problem of minimal cutset in an anisotropic network. We aim here to derive a large deviation principle for cutsets and deduce by contraction principle a lower large deviation principle for the maximal flow in $\Omega$.

For a given dimension d $\ge$ 2 and a finite measure $\nu$ on (0, +$\infty$), we consider $\xi$ a Poisson point process on R d x (0, +$\infty$) with intensity measure dc $\otimes$ $\nu$ where dc denotes the Lebesgue measure on R d. We consider the Boolean model $\Sigma$ = $\cup$ (c,r)$\in$$\xi$ B(c, r) where B(c, r) denotes the open ball centered at c with radius r. For every x, y $\in$ R d we define T (x, y) as the minimum time needed to travel from x to y by a traveler that walks at speed 1 outside $\Sigma$ and at infinite speed inside $\Sigma$. By a standard application of Kingman sub-additive theorem, one easily shows that T (0, x) behaves like $\mu$ x when x goes to infinity, where $\mu$ is a constant named the time constant in classical first passage percolation. In this paper we investigate the regularity of $\mu$ as a function of the measure $\nu$ associated with the underlying Boolean model.

We consider the standard first passage percolation model in the rescaled lattice $\mathbb{Z}^d$ for $d\geq 2$ and a bounded domain $\Omega$ in $\mathbb R ^d$. We denote by $\Gamma^1$ and $\Gamma^2$ two disjoint subsets of $\partial \Omega$ representing respectively the source and the sink, i.e., where the water can enter in $\Omega$ and escape from $\Omega$. A maximal stream is a vector measure $\overrightarrow{\mu}_n^{max}$ that describes how the maximal amount of fluid can enter through $\Gamma^1$ and spreads in $\Omega$. Under some assumptions on $\Omega$ and $G$, we already know a law of large number for $\overrightarrow{\mu}_n^{max}$. The sequence $(\overrightarrow{\mu}_n^{max})_{n\geq 1} $ converges almost surely to the set of solutions of a continuous deterministic problem of maximal stream in an anisotropic network. We aim here to derive a large deviation principle for streams and deduce by contraction principle the existence of a rate function for the upper large deviations of the maximal flow in $\Omega$.

The Teichm\"uller space $\mathcal{T}(\Sigma)$ of a surface $\Sigma$ is equipped with Thurston's asymmetric metric. Stretch lines are oriented geodesics for this metric on $\mathcal{T}(\Sigma)$. We give the asymptotic behavior of the lengths of the measured geodesic laminations as one follows a stretch line in the positive direction.

We consider the standard model of i.i.d. first passage percolation on Z^d given a distribution G on [0, +$\infty$] (including +$\infty$). We suppose that G({0}) > 1 -- p\_c(d), i.e., the edges of positive passage time are in the subcritical regime of percolation on Z^d. We consider a cylinder of basis an hyperrectangle of dimension d -- 1 whose sides have length n and of height h(n) with h(n) negligible compared to n (i.e., h(n)/n $\rightarrow$ 0 when n goes to infinity). We study the maximal flow from the top to the bottom of this cylinder. We already know that the maximal flow renormalized by n^(d--1) converges towards the flow constant which is null in the case G({0}) > 1 -- p\_c (d). The study of maximal flow is associated with the study of sets of edges of minimal capacity that cut the top from the bottom of the cylinder. If we denote by $\psi$\_n the minimal cardinal of such a set of edges, we prove here that $\psi$\_n /n^(d--1) converges almost surely towards a constant.

We consider the Boolean model on $\R^d$. We prove some equivalences between subcritical percolation properties. Let us introduce some notations to state one of these equivalences. Let $C$ denote the connected component of the origin in the Boolean model. Let $|C|$ denotes its volume. Let $\ell$ denote the maximal length of a chain of random balls from the origin. Under optimal integrability conditions on the radii, we prove that $E(|C|)$ is finite if and only if there exists $A,B >0$ such that $\P(\ell \ge n) \le Ae^{-Bn}$ for all $n \ge 1$.

We consider the model of i.i.d. first passage percolation on Z^d, where we associate with the edges of the graph a family of i.i.d. random variables with common distribution G on [0, +$\infty$] (including +$\infty$). Whereas the time constant is associated to the study of 1-dimensional paths with minimal weight, namely geodesics, the flow constant is associated to the study of (d--1)-dimensional surfaces with minimal weight. In this article, we investigate the existence of the flow constant under the only hypothesis that G({+$\infty$}) < p c (d) (in particular without any moment assumption), the convergence of some natural maximal flows towards this constant, and the continuity of this constant with regard to the distribution G.

We consider a non trivial Boolean model $\Sigma$ on ${\mathbb R}^d$ for $d\geq 2$. For every $x,y \in {\mathbb R}^d$ we define $T(x,y)$ as the minimum time needed to travel from $x$ to $y$ by a traveler that walks at speed $1$ outside $\Sigma$ and at infinite speed inside $\Sigma$. By a standard application of Kingman sub-additive theorem, one easily shows that $T(0,x)$ behaves like $\mu \|x\|$ when $\|x\|$ goes to infinity, where $\mu$ is a constant named the time constant in classical first passage percolation. In this paper we investigate the positivity of $\mu$. More precisely, under an almost optimal moment assumption on the radii of the balls of the Boolean model, we prove that $\mu\textgreater{}0$ if and only if the intensity $\lambda$ of the Boolean model satisfies $\lambda \textless{} \widehat{\lambda}\_c$, where $ \widehat{\lambda}\_c$ is one of the classical critical parameters defined in continuum percolation.

We consider two different objects on super-critical Bernoulli percolation on $\mathbb{Z}^d$ : the time constant for i.i.d. first-passage percolation (for $d\geq 2$) and the isoperimetric constant (for $d=2$). We prove that both objects are continuous with respect to the law of the environment. More precisely we prove that the isoperimetric constant of supercritical percolation in $\mathbb{Z}^2$ is continuous in the percolation parameter. As a corollary we prove that normalized sets achieving the isoperimetric constant are continuous with respect to the Hausdroff metric. Concerning first-passage percolation, equivalently we consider the model of i.i.d. first-passage percolation on $\mathbb{Z}^d$ with possibly infinite passage times: we associate with each edge $e$ of the graph a passage time $t(e)$ taking values in $[0,+\infty]$, such that $\mathbf{P}[t(e)<+\infty] >p_c(d)$. We prove the continuity of the time constant with respect to the law of the passage times. This extends the continuity property previously proved by Cox and Kesten for first passage percolation with finite passage times., Comment: 37 pages, 3 figures

We propose a new geometric approach to describe the qualitative dynamics of chemical reactions networks. By this method we identify metastable regimes, defined as low dimensional regions of the phase space close to which the dynamics is much slower compared to the rest of the phase space. Given the network topology and the orders of magnitude of kinetic parameters, the number of such metastable regimes is finite. The dynamics of the network can be described as a sequence of jumps from one metastable regime to another. We show that a geometrically computed connectivity graph restricts the set of possible jumps. We also provide finite state machine (Markov chain) models for such dynamic changes. Applied to signal transduction models, our approach unravels dynamical and functional capacities of signaling pathways, as well as parameters responsible for specificity of the pathway response. In particular, for a model of TGF$\beta$ signalling, we find that the ratio of TGFBR1 to TGFBR2 concentrations can be used to discriminate between metastable regimes. Using expression data from the NCI60 panel of human tumor cell lines, we show that aggressive and non-aggressive tumour cell lines function in different metastable regimes and can be distinguished by measuring the relative concentrations of receptors of the two types., Comment: arXiv admin note: text overlap with arXiv:1504.07833

The memoir Theorie der Parallellinien (1766) by Johann Heinrich Lambert is one of the founding texts of hyperbolic geometry, even though its author's aim was, like many of his pre-decessors', to prove that such a geometry does not exist. In fact, Lambert developed his theory with the hope of finding a contradiction in a geometry where all the Euclidean axioms are kept except the parallel axiom and that the latter is replaced by its negation. In doing so, he obtained several fundamental results of hyperbolic geometry. This was sixty years before the first writings of Lobachevsky and Bolyai appeared in print. In the present paper, we present Lambert's main results and we comment on them. A French translation of the Theorie der Parallellinien, together with an extensive commentary, has just appeared in print (A. Papadopoulos and G. Th{\'e}ret, La th{\'e}orie des lignes parall{\`e}les de Johann Heinrich Lambert. Collection Sciences dans l'Histoire, Librairie Scientifique et Technique Albert Blanchard, Paris, 2014).

We prove the continuity of the shape governing the asymptotic growth of the supercritical contact process in Z^d , with respect to the infection parameter. The proof is valid in any dimension d $\ge$ 1.

We show that the length function of a measured geodesic lamination is convex in Thurston's shear coordinates over Teichm\"uller space and strictly convex for generic laminations. We give some consequences of this result in the context of Thurston's asymmetric metric on Teichm\"uller space., Comment: 43 pages, 14 figures

We consider the model of i.i.d. first passage percolation on $\mathbb{Z}^d$ : we associate with each edge $e$ of the graph a passage time $t(e)$ taking values in $[0,+\infty]$, such that $\mathbb{P}[t(e)<+\infty] >p_c(d)$. Equivalently, we consider a standard (finite) i.i.d. first passage percolation model on a super-critical Bernoulli percolation performed independently. We prove a weak shape theorem without any moment assumption. We also prove that the corresponding time constant is positive if and only if $\mathbb{P}[t(e)=0]

The theory of geometric structures on a surface with nonempty boundary can be developed by using a decomposition of such a surface into hexagons, in the same way as the theory of geometric structures on a surface without boundary is developed using the decomposition of such a surface into pairs of pants. The basic elements of the theory for surfaces with boundary include the study of measured foliations and of hyperbolic structures on hexagons. It turns out that there is an interesting space of measured foliations on a hexagon, which is equipped with a piecewise-linear structure (in fact, a natural cell-decomposition), and this space is a natural boundary for the space of hyperbolic structures with geodesic boundary and right angles on such a hexagon. In this paper, we describe these spaces and the related structures.

We consider the standard first passage percolation model in the rescaled graph $\mathbb{Z}^d/n$ for $d\geq2$ and a domain $\Omega$ of boundary $\Gamma$ in $\mathbb{R}^d$. Let $\Gamma ^1$ and $\Gamma ^2$ be two disjoint open subsets of $\Gamma$, representing the parts of $\Gamma$ through which some water can enter and escape from $\Omega$. A law of large numbers for the maximal flow from $\Gamma ^1$ to $\Gamma ^2$ in $\Omega$ is already known. In this paper we investigate the asymptotic behavior of a maximal stream and a minimal cutset. A maximal stream is a vector measure $\vec{\mu}_n^{\max}$ that describes how the maximal amount of fluid can cross $\Omega$. Under conditions on the regularity of the domain and on the law of the capacities of the edges, we prove that the sequence $(\vec{\mu}_n^{\max})_{n\geq1}$ converges a.s. to the set of the solutions of a continuous deterministic problem of maximal stream in an anisotropic network. A minimal cutset can been seen as the boundary of a set $E_n^{\min}$ that separates $\Gamma ^1$ from $\Gamma ^2$ in $\Omega$ and whose random capacity is minimal. Under the same conditions, we prove that the sequence $(E_n^{\min})_{n\geq1}$ converges toward the set of the solutions of a continuous deterministic problem of minimal cutset. We deduce from this a continuous deterministic max-flow min-cut theorem and a new proof of the law of large numbers for the maximal flow. This proof is more natural than the existing one, since it relies on the study of maximal streams and minimal cutsets, which are the pertinent objects to look at., Comment: Published in at http://dx.doi.org/10.1214/13-AOP851 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We study the action of the elements of the mapping class group of a surface of finite type on the Teichm\"uller space of that surface equipped with Thurston's asymmetric metric. We classify such actions as elliptic, parabolic, hyperbolic and pseudo-hyperbolic, depending on whether the translation distance of such an element is zero or positive and whether the value of this translation distance is attained or not, and we relate these four types to Thurston's classification of mapping classes. The study is parallel to the one made by Bers in the setting of Teichm\"uller space equipped with Teichm\"uller's metric, and to the one made by Daskalopoulos and Wentworth in the setting of Teichm\"uller space equipped with the Weil-Petersson metric.

In the Teichm\"uller space of a hyperbolic surface of finite type, we construct geodesic lines for Thurston's asymmetric metric having the property that when they are traversed in the reverse direction, they are also geodesic lines (up to reparametrization). The lines we construct are special stretch lines in the sense of Thurston. They are directed by complete geodesic laminations that are not chain-recurrent, and they have a nice description in terms of Fenchel-Nielsen coordinates. At the basis of the construction are certain maps with controlled Lipschitz constants between right-angled hyperbolic hexagons having three non-consecutive edges of the same size. Using these maps, we obtain Lipschitz-minimizing maps between hyperbolic particular pairs of pants and, more generally, between some hyperbolic sufaces of finite type with arbitrary genus and arbitrary number of boundary components. The Lipschitz-minimizing maps that we contruct are distinct from Thurston's stretch maps.

Equip the edges of the lattice $\mathbb{Z}^2$ with i.i.d. random capacities. A law of large numbers is known for the maximal flow crossing a rectangle in $\mathbb{R}^2$ when the side lengths of the rectangle go to infinity. We prove that the lower large deviations are of surface order, and we prove the corresponding large deviation principle from below. This extends and improves previous large deviations results of Grimmett and Kesten (1984) obtained for boxes of particular orientation., Comment: 36 pages, 4 figures

We consider the standard first passage percolation model in the rescaled graph $\mathbb {Z}^d/n$ for $d\geq2$ and a domain $\Omega$ of boundary $\Gamma$ in $\mathbb {R}^d$. Let $\Gamma ^1$ and $\Gamma ^2$ be two disjoint open subsets of $\Gamma$ representing the parts of $\Gamma$ through which some water can enter and escape from $\Omega$. We investigate the asymptotic behavior of the flow $\phi_n$ through a discrete version $\Omega_n$ of $\Omega$ between the corresponding discrete sets $\Gamma ^1_n$ and $\Gamma ^2_n$. We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the upper large deviations of $\phi_n/n^{d-1}$ above a certain constant are of volume order, that is, decays exponentially fast with $n^d$. This article is part of a larger project in which the authors prove that this constant is the a.s. limit of $\phi_n/n^{d-1}$., Comment: Published in at http://dx.doi.org/10.1214/10-AAP732 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We consider the standard first passage percolation model in the rescaled graph $\mathbb{Z}^d/n$ for $d\geq 2$, and a domain $\Omega$ of boundary $\Gamma$ in $\mathbb{R}^d$. Let $\Gamma^1$ and $\Gamma^2$ be two disjoint open subsets of $\Gamma$, representing the parts of $\Gamma$ through which some water can enter and escape from $\Omega$. We investigate the asymptotic behaviour of the flow $\phi_n$ through a discrete version $\Omega_n$ of $\Omega$ between the corresponding discrete sets $\Gamma^1_n$ and $\Gamma^2_n$. We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the lower large deviations of $\phi_n/ n^{d-1}$ below a certain constant are of surface order., Comment: 23 pages, 8 figures

We consider the standard first passage percolation model in the rescaled graph $\mathbb{Z}^d/n$ for $d\geq 2$, and a domain $\Omega$ of boundary $\Gamma$ in $\mathbb{R}^d$. Let $\Gamma^1$ and $\Gamma^2$ be two disjoint open subsets of $\Gamma$, representing the parts of $\Gamma$ through which some water can enter and escape from $\Omega$. We investigate the asymptotic behaviour of the flow $\phi_n$ through a discrete version $\Omega_n$ of $\Omega$ between the corresponding discrete sets $\Gamma^1_n$ and $\Gamma^2_n$. We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, $\phi_n$ converges almost surely towards a constant $\phi_{\Omega}$, which is the solution of a continuous non-random min-cut problem. Moreover, we give a necessary and sufficient condition on the law of the capacity of the edges to ensure that $\phi_{\Omega} >0$., Comment: 41 pages, 8 figures

Equip the edges of the lattice $\mathbb{Z}^2$ with i.i.d. random capacities. We prove a law of large numbers for the maximal flow crossing a rectangle in $\mathbb{R}^2$ when the side lengths of the rectangle go to infinity. The value of the limit depends on the asymptotic behaviour of the ratio of the height of the cylinder over the length of its basis. This law of large numbers extends the law of large numbers obtained by Grimmett and Kesten (1984) for rectangles of particular orientation., Comment: 27 pages, 4 figures; minor modifications

A cylindrical stretch line is a stretch line, in the sense of Thurston, whose horocyclic lamination is a weighted multicurve. In this paper, we show that two correctly parameterized cylindrical lines are parallel if and only if these lines converge towards the same point in Thurston's boundary of Teichm\"uller space., Comment: 13 pages, 2 figures

We consider the standard first passage percolation model in $\ZZ^d$ for $d\geq 2$ and we study the maximal flow from the upper half part to the lower half part (respectively from the top to the bottom) of a cylinder whose basis is a hyperrectangle of sidelength proportional to $n$ and whose height is $h(n)$ for a certain height function $h$. We denote this maximal flow by $\tau_n$ (respectively $\phi_n$). We emphasize the fact that the cylinder may be tilted. We look at the probability that these flows, rescaled by the surface of the basis of the cylinder, are greater than $\nu(\vec{v})+\eps$ for some positive $\eps$, where $\nu(\vec{v})$ is the almost sure limit of the rescaled variable $\tau_n$ when $n$ goes to infinity. On one hand, we prove that the speed of decay of this probability in the case of the variable $\tau_n$ depends on the tail of the distribution of the capacities of the edges: it can decays exponentially fast with $n^{d-1}$, or with $n^{d-1} \min(n,h(n))$, or at an intermediate regime. On the other hand, we prove that this probability in the case of the variable $\phi_n$ decays exponentially fast with the volume of the cylinder as soon as the law of the capacity of the edges admits one exponential moment; the importance of this result is however limited by the fact that $\nu(\vec{v})$ is not in general the almost sure limit of the rescaled maximal flow $\phi_n$, but it is the case at least when the height $h(n)$ of the cylinder is negligible compared to $n$., Comment: 14 pages, 4 figures

We give a proof of an unpublished result of Thurston showing that given any hyperbolic metric on a surface of finite type with nonempty boundary, there exists another hyperbolic metric on the same surface for which the lengths of all simple closed geodesics are shorter. (This is not possible for surfaces of finite type with empty boundary.) Furthermore, we show that we can do the shortening in such a way that it is bounded below by a positive constant. This improves a recent result obtained by Parlier in [2]. We include this result in a discussion of the weak metric theory of the Teichm\"uller space of surfaces with nonempty boundary., Comment: Revised version, to appear in the Proceedings of the AMS

We consider some metrics and weak metrics defined on the Teichmueller space of a surface of finite type with nonempty boundary, that are defined using the hyperbolic length spectrum of simple closed curves and of properly embedded arcs, and we compare these metrics and weak metrics with the Teichm\"uller metric. The comparison is on subsets of Teichm\"uller space which we call "$\epsilon_0$-relative $\epsilon$-thick parts", and whose definition depends on the choice of some positive constants $\epsilon_0$ and $\epsilon$. Meanwhile, we give a formula for the Teichm\"uller metric of a surface with boundary in terms of extremal lengths of families of arcs., Comment: The revised version will appear in Monatshefte f\"ur Mathematik

We define and study metrics and weak metrics on the Teichmueller space of a surface of topologically finite type with boundary. These metrics and weak metrics are associated to the hyperbolic length spectrum of simple closed curves and of properly embedded arcs in the surface. We give a comparison between the defined metrics on regions of Teichmueller space which we call $\varepsilon_0$-relative $\epsilon$-thick parts} for $\epsilon >0$ and $\varepsilon_0\geq \epsilon>0$.

We consider the standard first passage percolation model in $\mathbb{Z}^d$ for $d\geq 2$. We are interested in two quantities, the maximal flow $\tau$ between the lower half and the upper half of the box, and the maximal flow $\phi$ between the top and the bottom of the box. A standard subadditive argument yields the law of large numbers for $\tau$ in rational directions. Kesten and Zhang have proved the law of large numbers for $\tau$ and $\phi$ when the sides of the box are parallel to the coordinate hyperplanes: the two variables grow linearly with the surface $s$ of the basis of the box, with the same deterministic speed. We study the probabilities that the rescaled variables $\tau /s$ and $\phi /s$ are abnormally small. For $\tau$, the box can have any orientation, whereas for $\phi$, we require either that the box is sufficiently flat, or that its sides are parallel to the coordinate hyperplanes. We show that these probabilities decay exponentially fast with $s$, when $s$ grows to infinity. Moreover, we prove an associated large deviation principle of speed $s$ for $\tau /s$ and $\phi /s$, and we improve the conditions required to obtain the law of large numbers for these variables., Comment: 39 pages, 4 figures; improvement of the moment conditions and introduction of new results in the revised version

Techniques for data-mining, latent semantic analysis, contextual search of databases, etc. have long ago been developed by computer scientists working on information retrieval (IR). Experimental scientists, from all disciplines, having to analyse large collections of raw experimental data (astronomical, physical, biological, etc.) have developed powerful methods for their statistical analysis and for clustering, categorising, and classifying objects. Finally, physicists have developed a theory of quantum measurement, unifying the logical, algebraic, and probabilistic aspects of queries into a single formalism. The purpose of this paper is twofold: first to show that when formulated at an abstract level, problems from IR, from statistical data analysis, and from physical measurement theories are very similar and hence can profitably be cross-fertilised, and, secondly, to propose a novel method of fuzzy hierarchical clustering, termed \textit{semantic distillation} -- strongly inspired from the theory of quantum measurement --, we developed to analyse raw data coming from various types of experiments on DNA arrays. We illustrate the method by analysing DNA arrays experiments and clustering the genes of the array according to their specificity., Comment: Accepted for publication in Studies in Computational Intelligence, Springer-Verlag

We consider the standard first passage percolation in $\mathbb{Z}^{d}$ for $d\geq 2$ and we denote by $\phi_{n^{d-1},h(n)}$ the maximal flow through the cylinder $]0,n]^{d-1} \times ]0,h(n)]$ from its bottom to its top. Kesten proved a law of large numbers for the maximal flow in dimension three: under some assumptions, $\phi_{n^{d-1},h(n)} / n^{d-1}$ converges towards a constant $\nu$. We look now at the probability that $\phi_{n^{d-1},h(n)} / n^{d-1}$ is greater than $\nu + \epsilon$ for some $\epsilon >0$, and we show under some assumptions that this probability decays exponentially fast with the volume of the cylinder. Moreover, we prove a large deviations principle for the sequence $(\phi_{n^{d-1},h(n)} / n^{d-1}, n\in \mathbb{N})$., Comment: 27 pages, 4 figures; small changes of notations

We consider the standard first passage percolation on $\mathbb{Z}^{d}$: with each edge of the lattice we associate a random capacity. We are interested in the maximal flow through a cylinder in this graph. Under some assumptions Kesten proved in 1987 a law of large numbers for the rescaled flow. Chayes and Chayes established that the large deviations far away below its typical value are of surface order, at least for the Bernoulli percolation and cylinders of certain height. Thanks to another approach we extend here their result to higher cylinders, and we transport this result to the model of first passage percolation., Comment: 11 pages, 2 figure; added references, changes on part 4.1