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Clément Cosco

Publications and source records attributed to Clément Cosco.

16 recordsLinked to original sources
arXiv2025

A central limit theorem for two-dimensional directed polymers with critical spatial correlation

On the 1+2 dimensional lattice, we consider a directed polymer in a random Gaussian environment that is independent in time and correlated in space. The spatial correlation is supposed to decay as $(\log |x|)^a /|x|^{2}$, $a>-1$, where the square in the polynomial is known to be critical (Lacoin, Ann. Prob. (2011)). We introduce an intermediate regime of temperature $β_N \propto \hat β/(\log N)^{\frac{a+2}{2}}$, under which the log-partition function $\log W_N^{β_N}$ converges in distribution towards a Gaussian random variable if $\hat β\in (0,\hat β_c)$, whereas $W_N^{β_N}$ vanishes for $\hat β\geq \hat β_c$. The variance of the limiting Gaussian distribution, which is given by an inverse Bessel function, is determined by an induction scheme whose multi-scale dependence reflects the critical nature of the correlation. The Gaussianity of the limit follows from a decoupling argument of Cosco, Donadini (2024+).

math.PR↗
arXiv2025

On the Central Limit Theorem for the log-partition function of 2D directed polymers

The log-partition function $ \log W_N(β)$ of the two-dimensional directed polymer in random environment is known to converge in distribution to a normal distribution when considering temperature in the subcritical regime $β=β_N=\hatβ\sqrt{π/\log N}$, $\hatβ\in (0,1)$ (Caravenna, Sun, Zygouras, Ann. Appl. Prob. (2017)). In this paper, we present an elementary proof of this result relying on a decoupling argument and the central limit theorem for sums of independent random variables. The argument is inspired by an analogy of the model to branching random walks.

math.PR↗
arXiv2025

High moments of 2d directed polymers up to quasi-criticality

We consider two-dimensional directed polymers in random environment in the sub-critical regime and in the quasi-critical regime introduced recently by Caravenna, Cottini and Rossi, arXiv:2307.02453v1. For $q\leq q_N$ with $q_N\to\infty$ diverging at a suitable rate with the size of the system, we obtain upper bound estimates on the $q$-moment of the partition function for general environments. In the sub-critical regime, our results improve the $q_N$-threshold obtained for Gaussian environment in Cosco, Zeitouni, Comm. Math. Phys (2023). As a corollary, we derive large deviation estimates with a Gaussian rate function.

math.PR↗
arXiv2025

The maximum of the two dimensional Gaussian directed polymer in the subcritical regime

We study the maximum $ϕ_N^*$ of the partition function of the two dimensional (subcritical) Gaussian directed polymer over an $\sqrt N \times \sqrt N$ box. We show that $ϕ_N^*/\log N$ converges towards a constant $σ^*$, which we identify to be the same as for the maximum of a branching random walk with a slowly varying variance profile as studied in Fang-Zeitouni, J. Stat. Phys. 2012 and (in the context of the generalized random energy model) in Bovier-Kurkova, Ann. Inst. H. Poincare 2004.

math.PR↗
arXiv2023

Brownian Polymers in Poissonian Environment: a survey

We consider a space-time continuous directed polymer in random environment. The path is Brownian and the medium is Poissonian. We review many results obtained in the last decade, and also we present new ones. In this fundamental setup, we can make use of fine formulas and strong tools from stochastic analysis for Gaussian or Poisson measure, together with martingale techniques. These notes cover the matter of a course presented during the Jean-Morlet chair 2017 of CIRM "Random Structures in Statistical Mechanics and Mathematical Physics" in Marseille.

math.PR↗
arXiv2023

Moments of partition functions of 2D Gaussian polymers in the weak disorder regime -- I

Let $W_N(β) = \mathrm{E}_0\left[e^{ \sum_{n=1}^N βω(n,S_n) - Nβ^2/2}\right]$ be the partition function of a two-dimensional directed polymer in a random environment, where $ω(i,x), i\in \mathbb{Z}_+, x\in \mathbb{Z}^2$ are i.i.d.\ standard normal and $\{S_n\}$ is the path of a random walk. With $β=β_N=\hatβ\sqrt{π/\log N}$ and $\hat β\in (0,1)$ (the subcritical window), $\log W_N(β_N)$ is known to converge in distribution to a Gaussian law of mean $-λ^2/2$ and variance $λ^2$, with $λ^2=\log \big(1/(1-\hatβ^2\big)$ (Caravenna, Sun, Zygouras, Ann. Appl. Probab. (2017)). We study in this paper the moments $\mathbb{E} [W_N( β_N)^q]$ in the subcritical window, for $q=O(\sqrt{\log N})$. The analysis is based on ruling out triple intersections

math.PR↗
arXiv2023

Asymptotics of the $p$-capacity in the critical regime

In this note, we are interested in the asymptotics as $n\to\infty$ of the $p$-capacity between the origin and the set $nB$, where $B$ is the boundary of the unit ball of the lattice $\mathbb Z^d$. The $p$-capacity is defined as the minimum of the Dirichlet energy $\frac{1}{2}\sum_{x\in \mathbb Z^d} \sum_{y\sim x} |f(x)-f(y)|^{p}$ with $f$ subject to the boundary conditions $f(0)=0$ and $f\geq 1$ on $nB$. This variational problem has arisen in particular in the study of large deviations for first passage percolation. For $p d$ the capacity vanishes polynomially fast. The present paper deals with the case $p=d$, for which we prove that the $p$-capacity vanishes as $c_d (\log n)^{-d+1}$ with an explicit constant $c_d$.

math.AP↗
arXiv2023

Moments of partition functions of 2D Gaussian polymers in the weak disorder regime -- II

Let $W_N(β) = \mathrm{E}_0\left[e^{ \sum_{n=1}^N βω(n,S_n) - Nβ^2/2}\right]$ be the partition function of a two-dimensional directed polymer in a random environment, where $ω(i,x), i\in \mathbb{N}, x\in \mathbb{Z}^2$ are i.i.d. standard normal and $\{S_n\}$ is the path of a random walk. With $β=β_N=\widehatβ \sqrt{π/\log N}$ and $\widehatβ\in (0,1)$ (the subcritical window), $\log W_N(β_N)$ is known to converge in distribution to a Gaussian law of mean $-λ^2/2$ and variance $λ^2$, with $λ^2=\log ((1-\widehatβ^2)^{-1})$ (Caravenna, Sun, Zygouras, Ann. Appl. Probab. (2017)). We study in this paper the moments $\mathbb E [W_N( β_N)^q]$ in the subcritical window, and prove a lower bound that matches for $q=O(\sqrt{\log N})$ the upper bound derived by us in Cosco, Zeitouni, arXiv:2112.03767 [math.PR]. The analysis is based on appropriate decouplings and a Poisson convergence that uses the method of ''two moments suffice''.

math.PR↗
arXiv2023

A variational formula for large deviations in First-passage percolation under tail estimates

Consider first passage percolation with identical and independent weight distributions and first passage time ${\rm T}$. In this paper, we study the upper tail large deviations $\mathbb{P}({\rm T}(0,nx)>n(μ+ξ))$, for $ξ>0$ and $x\neq 0$ with a time constant $μ$ and a dimension $d$, for weights that satisfy a tail assumption $ β_1\exp{(-αt^r)}\leq \mathbb P(τ_e>t)\leq β_2\exp{(-αt^r)}.$ When $r\leq 1$ (this includes the well-known Eden growth model), we show that the upper tail large deviation decays as $\exp{(-(2dξ+o(1))n)}$. When $1< r\leq d$, we find that the rate function can be naturally described by a variational formula, called the discrete p-Capacity, and we study its asymptotics. For $r n(μ+ξ)$ is described by a localization of high weights around the origin. The picture changes for $r\geq d$ where the configuration is not anymore localized.

math.PR↗
arXiv2021

Space-time fluctuation of the Kardar-Parisi-Zhang equation in $d\geq 3$ and the Gaussian free field

We study the solution $h_\varepsilon$ of the Kardar-Parisi-Zhang (KPZ) equation for $d \geq 3$: $$ \frac{\partial}{\partial t} h_{\varepsilon} = \frac12 Δh_{\varepsilon} + \bigg[\frac12 |\nabla h_\varepsilon |^2 - C_\varepsilon\bigg]+ β\varepsilon^{\frac{d-2}2} ξ_{\varepsilon} $$ with $h_\varepsilon(0,x)=0$. Here $ξ_\varepsilon=ξ\star ϕ_\varepsilon$ is a spatially smoothened (at scale $\varepsilon$) Gaussian space-time white noise and $C_\varepsilon$ is a divergent constant as $\varepsilon\to 0$. When the disorder $β$ is sufficiently small and $\varepsilon\to 0$, $h_\varepsilon(t,x)- \mathfrak h^{\mathrm{st}}_{\varepsilon}(t,x)\to 0$ in probability where $\mathfrak h^{\mathrm{st}}_{\varepsilon}(t,x)$ is the {\emph stationary solution} of the KPZ equation - more precisely, $\mathfrak h^{\mathrm{st}}_{\varepsilon}$solves the above equation with a random initial condition (that is independent of the driving noise $ξ$) and its law is constant in $(\varepsilon,t,x)$. In the present article we quantify the rate of the above convergence in this regime and show that the fluctuation {\emph about} the stationary solution $$ (\varepsilon^{1-\frac d2} [h_\varepsilon(t,x) - \mathfrak h^{\mathrm{st}}_{\varepsilon}(t,x)])_{x,t} $$ converges pointwise (with finite dimensional distributions in space and time) to a Gaussian free field (GFF) evolved by the deterministic heat equation. We also identify the fluctuations {\it of} the stationary solution itself and show that the rescaled averages $\int_{\mathbb R^d} {\mathrm d} x φ(x) \varepsilon^{1-\frac d2} [\mathfrak h^{\mathrm{st}}_{\varepsilon}(t,x)- \mathbb E(\mathfrak h^{\mathrm{st}}_{\varepsilon}(t,x))]$ converge to that of the {\emph stationary solution} of the stochastic heat equation with additive noise, but with (random) {\emph GFF marginals} (instead of flat initial condition).

math.PR↗
arXiv2020

Gaussian fluctuations for the directed polymer partition function for $d\geq 3$ and in the whole $L^2$-region

We consider the discrete directed polymer model with i.i.d. environment and we study the fluctuations of the tail $n^{(d-2)/4}(W_\infty - W_n)$ of the normalized partition function. It was proven by Comets and Liu, that for sufficiently high temperature, the fluctuations converge in distribution towards the product of the limiting partition function and an independent Gaussian random variable. We extend the result to the whole $L^2$-region, which is predicted to be the maximal high-temperature region where the Gaussian fluctuations should occur under the considered scaling. To do so, we manage to avoid the heavy 4th-moment computation and instead rely on the local limit theorem for polymers and homogenization.

math.PR↗
arXiv2020

Law of large numbers and fluctuations in the sub-critical and $L^2$ regions for SHE and KPZ equation in dimension $d\geq 3$

There have been recently several works studying the regularized stochastic heat equation (SHE) and Kardar-Parisi-Zhang (KPZ) equation in dimension $d\geq 3$ as the smoothing parameter is switched off, but most of the results did not hold in the full temperature regions where they should. Inspired by martingale techniques coming from the directed polymers literature, we first extend the law of large numbers for SHE obtained in [MSZ16] to the full weak disorder region of the associated polymer model and to more general initial conditions. We further extend the Edwards-Wilkinson regime of the SHE and KPZ equation studied in [GRZ18,MU17,DGRZ20] to the full $L^2$-region, along with multidimensional convergence and general initial conditions for the KPZ equation (and SHE), which were not proven before. To do so, we rely on a martingale CLT combined with a refinement of the local limit theorem for polymers.

math.PR↗
arXiv2020

Topologically induced metastability in periodic XY chain

Non-trivial topological behavior appears in many different contexts in statistical physics, perhaps the most known one being the Kosterlitz-Thouless phase transition in the two dimensional XY model. We study the behavior of a simpler, one dimensional, XY chain with periodic boundary and strong interactions; but rather than concentrating on the equilibrium measure we try to understand its dynamics. The equivalent of the Kosterlitz-Thouless transition in this one dimensional case happens when the interaction strength scales like the size of the system $N$, yet we show that a sharp transition for the dynamics occurs at the scale of $\log N$ -- when the interactions are weaker than a certain threshold topological phases could not be observed over long times, while for interactions that are stronger than that threshold topological phases become metastable, surviving for diverging time scales.

math.PR↗
arXiv2019

Fluctuation and Rate of Convergence for the Stochastic Heat Equation in Weak Disorder

We consider the stochastic heat equation on $\mathbb R^d$ with multiplicative space-time white noise noise smoothed in space. For $d\geq 3$ and small noise intensity, the solution is known to converge to a strictly positive random variable as the smoothing parameter vanishes. In this regime, we study the rate of convergence and show that the pointwise fluctuations of the smoothened solutions as well as that of the underlying martingale of the Brownian directed polymer converge to a Gaussian limit.

math.PR↗
arXiv2018

The Intermediate Disorder Regime for Brownian Directed Polymers in Poisson Environment

We consider the Brownian directed polymer in Poissonian environment in dimension 1+1, under the so-called intermediate disorder regime, which is a crossover regime between the strong and weak disorder regions. We show that, under a diffusive scaling involving different parameters of the system, the renormalized point-to-point partition function of the polymer converges in law to the solution of the stochastic heat equation with Gaussian multiplicative noise. The Poissonian environment provides a natural setting and strong tools, such as the Wiener-Ito chaos expansion, which, applied to the partition function, is the basic ingredient of the proof.

math.PR↗