arXiv subjects

Pietro Caputo

Publications and source records attributed to Pietro Caputo.

At least 55 records · Page 3Linked to original sources

arXiv2017

Entropic repulsion in $|\nabla ϕ|^p$ surfaces: a large deviation bound for all $p\geq 1$

We consider the $(2+1)$-dimensional generalized solid-on-solid (SOS) model, that is the random discrete surface with a gradient potential of the form $|\nablaϕ|^{p}$, where $p\in [1,+\infty]$. We show that at low temperature, for a square region $Λ$ with side $L$, both under the infinite volume measure and under the measure with zero boundary conditions around $Λ$, the probability that the surface is nonnegative in $Λ$ behaves like $\exp(-4βτ_{p,β} L H_p(L) )$, where $β$ is the inverse temperature, $τ_{p,β}$ is the surface tension at zero tilt, or step free energy, and $H_p(L)$ is the entropic repulsion height, that is the typical height of the field when a positivity constraint is imposed. This generalizes recent results obtained in \cite{CMT} for the standard SOS model ($p=1$).

math.PR↗

arXiv2016

Entropy production in nonlinear recombination models

We study the convergence to equilibrium of a class of nonlinear recombination models. In analogy with Boltzmann's H theorem from kinetic theory, and in contrast with previous analysis of these models, convergence is measured in terms of relative entropy. The problem is formulated within a general framework that we refer to as Reversible Quadratic Systems. Our main result is a tight quantitative estimate for the entropy production functional. Along the way we establish some new entropy inequalities generalizing Shearer's and related inequalities.

math.PR↗

arXiv2016

Large deviations of empirical neighborhood distribution in sparse random graphs

Consider the Erdős-Renyi random graph on n vertices where each edge is present independently with probability c/n, with c>0 fixed. For large n, a typical random graph locally behaves like a Galton-Watson tree with Poisson offspring distribution with mean c. Here, we study large deviations from this typical behavior within the framework of the local weak convergence of finite graph sequences. The associated rate function is expressed in terms of an entropy functional on unimodular measures and takes finite values only at measures supported on trees. We also establish large deviations for other commonly studied random graph ensembles such as the uniform random graph with given number of edges growing linearly with the number of vertices, or the uniform random graph with given degree sequence. To prove our results, we introduce a new configuration model which allows one to sample uniform random graphs with a given neighborhood distribution, provided the latter is supported on trees. We also introduce a new class of unimodular random trees, which generalizes the usual Galton Watson tree with given degree distribution to the case of neighborhoods of arbitrary finite depth. These generalized Galton Watson trees turn out to be useful in the analysis of unimodular random trees and may be considered to be of interest in their own right.

math.PR↗

arXiv2015

On the probability of staying above a wall for the (2+1)-dimensional SOS model at low temperature

We obtain sharp asymptotics for the probability that the (2+1)-dimensional discrete SOS interface at low temperature is positive in a large region. For a square region $Λ$, both under the infinite volume measure and under the measure with zero boundary conditions around $Λ$, this probability turns out to behave like $\exp(-τ_β(0) L \log L )$, with $τ_β(0)$ the surface tension at zero tilt, also called step free energy, and $L$ the box side. This behavior is qualitatively different from the one found for continuous height massless gradient interface models.

math.PR↗

arXiv2015

Random lattice triangulations: Structure and algorithms

The paper concerns lattice triangulations, that is, triangulations of the integer points in a polygon in $\mathbb{R}^2$ whose vertices are also integer points. Lattice triangulations have been studied extensively both as geometric objects in their own right and by virtue of applications in algebraic geometry. Our focus is on random triangulations in which a triangulation $σ$ has weight $λ^{|σ|}$, where $λ$ is a positive real parameter, and $|σ|$ is the total length of the edges in $σ$. Empirically, this model exhibits a "phase transition" at $λ=1$ (corresponding to the uniform distribution): for $λ<1$ distant edges behave essentially independently, while for $λ>1$ very large regions of aligned edges appear. We substantiate this picture as follows. For $λ<1$ sufficiently small, we show that correlations between edges decay exponentially with distance (suitably defined), and also that the Glauber dynamics (a local Markov chain based on flipping edges) is rapidly mixing (in time polynomial in the number of edges in the triangulation). This dynamics has been proposed by several authors as an algorithm for generating random triangulations. By contrast, for $λ>1$ we show that the mixing time is exponential. These are apparently the first rigorous quantitative results on the structure and dynamics of random lattice triangulations.

math.PR↗

arXiv2015

Dynamics of Lattice Triangulations on Thin Rectangles

We consider random lattice triangulations of $n\times k$ rectangular regions with weight $λ^{|σ|}$ where $λ>0$ is a parameter and $|σ|$ denotes the total edge length of the triangulation. When $λ\in(0,1)$ and $k$ is fixed, we prove a tight upper bound of order $n^2$ for the mixing time of the edge-flip Glauber dynamics. Combined with the previously known lower bound of order $\exp(Ω(n^2))$ for $λ>1$ [3], this establishes the existence of a dynamical phase transition for thin rectangles with critical point at $λ=1$.

math.PR↗

arXiv2015

Convergence to equilibrium for a directed (1+d)-dimensional polymer

We consider a flip dynamics for directed (1+d)-dimensional lattice paths with length L. The model can be interpreted as a higher dimensional version of the simple exclusion process, the latter corresponding to the case d=1. We prove that the mixing time of the associated Markov chain scales like L^2\log L up to a d-dependent multiplicative constant. The key step in the proof of the upper bound is to show that the system satisfies a logarithmic Sobolev inequality on the diffusive scale L^2 for every fixed d, which we achieve by a suitable induction over the dimension together with an estimate for adjacent transpositions. The lower bound is obtained with a version of Wilson's argument for the one-dimensional case.

math.PR↗

arXiv2015

Approximate tensorization of entropy at high temperature

We show that for weakly dependent random variables the relative entropy functional satisfies an approximate version of the standard tensorization property which holds in the independent case. As a corollary we obtain a family of dimensionless logarithmic Sobolev inequalities. In the context of spin systems on a graph, the weak dependence requirements resemble the well known Dobrushin uniqueness conditions. Our results can be considered as a discrete counterpart of a recent work of Katalin Marton. We also discuss some natural generalizations such as approximate Shearer estimates and subadditivity of entropy.

math.PR↗

arXiv2014

A large deviation principle for Wigner matrices without Gaussian tails

We consider $n\times n$ Hermitian matrices with i.i.d. entries $X_{ij}$ whose tail probabilities $\mathbb {P}(|X_{ij}|\geq t)$ behave like $e^{-at^α}$ for some $a>0$ and $α\in(0,2)$. We establish a large deviation principle for the empirical spectral measure of $X/\sqrt{n}$ with speed $n^{1+α/2}$ with a good rate function $J(μ)$ that is finite only if $μ$ is of the form $μ=μ_{\mathrm{sc}}\boxplusν$ for some probability measure $ν$ on $\mathbb {R}$, where $\boxplus$ denotes the free convolution and $μ_{\mathrm{sc}}$ is Wigner's semicircle law. We obtain explicit expressions for $J(μ_{\mathrm{sc}}\boxplusν)$ in terms of the $α$th moment of $ν$. The proof is based on the analysis of large deviations for the empirical distribution of very sparse random rooted networks.

math.PR↗

arXiv2014

Multi-level pinning problems for random walks and self-avoiding lattice paths

We consider a generalization of the classical pinning problem for integer-valued random walks conditioned to stay non-negative. More specifically, we take pinning potentials of the form $\sum_{j\geq 0}ε_j N_j$, where $N_j$ is the number of visits to the state $j$ and $\{ε_j\}$ is a non-negative sequence. Partly motivated by similar problems for low-temperature contour models in statistical physics, we aim at finding a sharp characterization of the threshold of the wetting transition, especially in the regime where the variance $σ^2$ of the single step of the random walk is small. Our main result says that, for natural choices of the pinning sequence $\{ε_j\}$, localization (respectively delocalization) occurs if $σ^{-2}\sum_{ j\geq0}(j+1)ε_j\geqδ^{-1}$ (respectively $\le δ$), for some universal $δ<1$. Our finding is reminiscent of the classical Bargmann-Jost-Pais criteria for the absence of bound states for the radial Schrödinger equation. The core of the proof is a recursive argument to bound the free energy of the model. Our approach is rather robust, which allows us to obtain similar results in the case where the random walk trajectory is replaced by a self-avoiding path $γ$ in $\mathbb Z^2$ with weight $\exp(-β|γ|)$, $|γ|$ being the length of the path and $β>0$ a large enough parameter. This generalization is directly relevant for applications to the above mentioned contour models.

math.PR↗

arXiv2014

Dynamics of $(2+1)$-dimensional SOS surfaces above a wall: Slow mixing induced by entropic repulsion

We study the Glauber dynamics for the $(2+1)\mathrm{D}$ Solid-On-Solid model above a hard wall and below a far away ceiling, on an $L\times L$ box of $\mathbb{Z}^2$ with zero boundary conditions, at large inverse-temperature $β$. It was shown by Bricmont, El Mellouki and Fröhlich [J. Stat. Phys. 42 (1986) 743-798] that the floor constraint induces an entropic repulsion effect which lifts the surface to an average height $H\asymp(1/β)\log L$. As an essential step in understanding the effect of entropic repulsion on the Glauber dynamics we determine the equilibrium height $H$ to within an additive constant: $H=(1/4β)\log L+O(1)$. We then show that starting from zero initial conditions the surface rises to its final height $H$ through a sequence of metastable transitions between consecutive levels. The time for a transition from height $h=aH$, $a\in(0,1)$, to height $h+1$ is roughly $\exp(cL^a)$ for some constant $c>0$. In particular, the mixing time of the dynamics is exponentially large in $L$, that is, $T_{\mathrm{MIX}}\geq e^{cL}$. We also provide the matching upper bound $T_{\mathrm{MIX}}\leq e^{c'L}$, requiring a challenging analysis of the statistics of height contours at low temperature and new coupling ideas and techniques. Finally, to emphasize the role of entropic repulsion we show that without a floor constraint at height zero the mixing time is no longer exponentially large in $L$.

math.PR↗

arXiv2014

Spectrum of Markov generators on sparse random graphs

We investigate the spectrum of the infinitesimal generator of the continuous time random walk on a randomly weighted oriented graph. This is the non-Hermitian random nxn matrix L defined by L(j,k)=X(j,k) if k<>j and L(j,j)=-sum(L(j,k),k<>j), where X(j,k) are i.i.d. random weights. Under mild assumptions on the law of the weights, we establish convergence as n tends to infinity of the empirical spectral distribution of L after centering and rescaling. In particular, our assumptions include sparse random graphs such as the oriented Erdös-Rényi graph where each edge is present independently with probability p(n)->0 as long as np(n) >> (log(n))^6. The limiting distribution is characterized as an additive Gaussian deformation of the standard circular law. In free probability terms, this coincides with the Brown measure of the free sum of the circular element and a normal operator with Gaussian spectral measure. The density of the limiting distribution is analyzed using a subordination formula. Furthermore, we study the convergence of the invariant measure of L to the uniform distribution and establish estimates on the extremal eigenvalues of L.

math.PR↗

arXiv2013

Scaling limit and cube-root fluctuations in SOS surfaces above a wall

Consider the classical $(2+1)$-dimensional Solid-On-Solid model above a hard wall on an $L\times L$ box of $\bbZ^2$. The model describes a crystal surface by assigning a non-negative integer height $η_x$ to each site $x$ in the box and 0 heights to its boundary. The probability of a surface configuration $η$ is proportional to $\exp(-β\mathcal{H}(η))$, where $β$ is the inverse-temperature and $\mathcal{H}(η)$ sums the absolute values of height differences between neighboring sites. We give a full description of the shape of the SOS surface for low enough temperatures. First we show that with high probability the height of almost all sites is concentrated on two levels, $H(L)=\lfloor (1/4β)\log L\rfloor$ and $H(L)-1$. Moreover, for most values of $L$ the height is concentrated on the single value $H(L)$. Next, we study the ensemble of level lines corresponding to the heights $(H(L),H(L)-1,...)$. We prove that w.h.p. there is a unique macroscopic level line for each height. Furthermore, when taking a diverging sequence of system sizes $L_k$, the rescaled macroscopic level line at height $H(L_k)-n$ has a limiting shape if the fractional parts of $(1/4β)\log L_k$ converge to a noncritical value. The scaling limit is an explicit convex subset of the unit square $Q$ and its boundary has a flat component on the boundary of $Q$. Finally, the highest macroscopic level line has $L_k^{1/3+o(1)}$ fluctuations along the flat part of the boundary of its limiting shape.

math.PR↗

arXiv2012

Spectrum of large random reversible Markov chains: Heavy-tailed weights on the complete graph

We consider the random reversible Markov kernel K obtained by assigning i.i.d. nonnegative weights to the edges of the complete graph over n vertices and normalizing by the corresponding row sum. The weights are assumed to be in the domain of attraction of an $α$-stable law, $α\in(0,2)$. When $1\leqα<2$, we show that for a suitable regularly varying sequence $κ_n$ of index $1-1/α$, the limiting spectral distribution $μ_α$ of $κ_nK$ coincides with the one of the random symmetric matrix of the un-normalized weights (Lévy matrix with i.i.d. entries). In contrast, when $0<α<1$, we show that the empirical spectral distribution of K converges without rescaling to a nontrivial law $\widetildeμ_α$ supported on [-1,1], whose moments are the return probabilities of the random walk on the Poisson weighted infinite tree (PWIT) introduced by Aldous. The limiting spectral distributions are given by the expected value of the random spectral measure at the root of suitable self-adjoint operators defined on the PWIT. This characterization is used together with recursive relations on the tree to derive some properties of $μ_α$ and $\widetildeμ_α$. We also study the limiting behavior of the invariant probability measure of K.

math.PR↗

arXiv2012

The shape of the $(2+1)$D SOS surface above a wall

We give a full description for the shape of the classical (2+1)\Dim Solid-On-Solid model above a wall, introduced by Temperley (1952). On an $L\times L$ box at a large inverse-temperature $β$ the height of most sites concentrates on a single level $h = \lfloor (1/4β)\log L\rfloor$ for most values of $L$. For a sequence of diverging boxes the ensemble of level lines of heights $(h,h-1,...)$ has a scaling limit in Hausdorff distance iff the fractional parts of $(1/4β)\log L$ converge to a noncritical value. The scaling limit is explicitly given by nested distinct loops formed via translates of Wulff shapes. Finally, the $h$-level lines feature $L^{1/3+o(1)}$ fluctuations from the side boundaries.

math.PR↗

arXiv2011

Spectrum of non-Hermitian heavy tailed random matrices

Let (X_{jk})_{j,k>=1} be i.i.d. complex random variables such that |X_{jk}| is in the domain of attraction of an alpha-stable law, with 0< alpha <2. Our main result is a heavy tailed counterpart of Girko's circular law. Namely, under some additional smoothness assumptions on the law of X_{jk}, we prove that there exists a deterministic sequence a_n ~ n^{1/alpha} and a probability measure mu_alpha on C depending only on alpha such that with probability one, the empirical distribution of the eigenvalues of the rescaled matrix a_n^{-1} (X_{jk})_{1<=j,k<=n} converges weakly to mu_alpha as n tends to infinity. Our approach combines Aldous & Steele's objective method with Girko's Hermitization using logarithmic potentials. The underlying limiting object is defined on a bipartized version of Aldous' Poisson Weighted Infinite Tree. Recursive relations on the tree provide some properties of mu_alpha. In contrast with the Hermitian case, we find that mu_alpha is not heavy tailed.

math.PR↗

arXiv2011

Mixing times of monotone surfaces and SOS interfaces: a mean curvature approach

We consider stochastic spin-flip dynamics for: (i) monotone discrete surfaces in Z^3 with planar boundary height and (ii) the one-dimensional discrete Solid-on-Solid (SOS) model confined to a box. In both cases we show almost optimal bounds O(L^2polylog(L)) for the mixing time of the chain, where L is the natural size of the system. The dynamics at a macroscopic scale should be described by a deterministic mean curvature motion such that each point of the surface feels a drift which tends to minimize the local surface tension. Inspired by this heuristics, our approach consists in bounding the dynamics with an auxiliary one which, with very high probability, follows quite closely the deterministic mean curvature evolution. Key technical ingredients are monotonicity, coupling and an argument due to D.B. Wilson in the framework of lozenge tiling Markov Chains. Our approach works equally well for both models despite the fact that their equilibrium maximal height fluctuations occur on very different scales (logarithmic for monotone surfaces and L^{1/2} for the SOS model). Finally, combining techniques from kinetically constrained spin systems together with the above mixing time result, we prove an almost diffusive lower bound of order 1/L^2 up to logarithmic corrections for the spectral gap of the SOS model with horizontal size L and unbounded heights.

math.PR↗

arXiv2010

"Zero" temperature stochastic 3D Ising model and dimer covering fluctuations: a first step towards interface mean curvature motion

We consider the Glauber dynamics for the Ising model with "+" boundary conditions, at zero temperature or at temperature which goes to zero with the system size (hence the quotation marks in the title). In dimension d=3 we prove that an initial domain of linear size L of "-" spins disappears within a time τ_+ which is at most L^2(\log L)^c and at least L^2/(c\log L), for some c>0. The proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the motion by mean curvature that is expected to describe, on large time-scales, the evolution of the interface between "+" and "-" domains. The analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factor, is the first rigorous confirmation of the expected behavior τ_+\simeq const\times L^2, conjectured on heuristic grounds. In dimension d=2, τ_+ can be shown to be of order L^2 without logarithmic corrections: the upper bound was proven in [Fontes, Schonmann, Sidoravicius, 2002] and here we provide the lower bound. For d=2, we also prove that the spectral gap of the generator behaves like c/L for L large, as conjectured in [Bodineau-Martinelli, 2002].

math-ph↗