arXiv subjects

Jeremy Quastel

Publications and source records attributed to Jeremy Quastel.

47 records · Page 3Linked to original sources

arXiv2010

The Intermediate Disorder Regime for Directed Polymers in Dimension 1+1

We introduce a new disorder regime for directed polymers with one space and one time dimension that is accessed by scaling the inverse temperature parameter βwith the length of the polymer n. We scale β_n := βn^{-α} for alpha non-negative. This scaling sits in between the usual weak disorder (β= 0) and strong disorder regimes (β> 0). The fluctuation exponents zeta for the polymer endpoint and χfor the free energy depend on αin this regime, with α= 0 corresponding to the usual polymer exponents ζ= 2/3, χ= 1/3 and α>= 1/4 corresponding to the simple random walk exponents ζ= 1/2, χ= 0. For 0 < α< 1/4 the exponents interpolate linearly between these two extremes. At α= 1/4 we exactly identify the limiting distribution of the free energy and the end point of the polymer.

cond-mat.stat-mech↗

arXiv2009

Scaling exponent for the Hopf-Cole solution of KPZ/Stochastic Burgers

We consider the stochastic heat equation $\partial_tZ= \partial_x^2 Z - Z \dot W$ on the real line, where $\dot W$ is space-time white noise. $h(t,x)=-\log Z(t,x)$ is interpreted as a solution of the KPZ equation, and $u(t,x)=\partial_x h(t,x)$ as a solution of the stochastic Burgers equation. We take $Z(0,x)=\exp\{B(x)\}$ where $B(x)$ is a two-sided Brownian motion, corresponding to the stationary solution of the stochastic Burgers equation. We show that there exist $0< c_1\le c_2 <\infty$ such that $c_1t^{2/3}\le \Var (\log Z(t,x))\le c_2 t^{2/3}.$ Analogous results are obtained for some moments of the correlation functions of $u(t,x)$. In particular, it is shown that the excess diffusivity satisfies $c_1t^{1/3}\le D(t) \le c_2 t^{1/3}.$ The proof uses approximation by weakly asymmetric simple exclusion processes, for which we obtain the microscopic analogies of the results by coupling.

math.PR↗

arXiv2009

Effect of Noise on Front Propagation in Reaction-Diffusion equations of KPP type

We consider reaction-diffusion equations of KPP type in one spatial dimension, perturbed by a Fisher-Wright white noise, under the assumption of uniqueness in distribution. Examples include the randomly perturbed Fisher-KPP equations $ \partial_t u = \partial_x^2 u + u(1-u) + ε\sqrt{u(1-u)}\dot W, $ and $ \partial_t u = \partial_x^2 u + u(1-u) + ε\sqrt{u}\dot W, $ where $\dot W= \dot W(t,x)$ is a space-time white noise. We prove the Brunet-Derrida conjecture that the speed of traveling fronts is asymptotically $ 2-π^2 |\log ε^2|^{-2} $ up to a factor of order $ (\log|\logε|)|\logε|^{-3}$.

math.PR↗

arXiv2007

A note on the diffusivity of finite-range asymmetric exclusion processes on Z

The diffusivity $D(t)$ of finite-range asymmetric exclusion processes on $\mathbb Z$ with non-zero drift is expected to be of order $t^{1/3}$. Seppäläinen and Balázs recently proved this conjecture for the nearest neighbor case. We extend their results to general finite range exclusion by proving that the Laplace transform of the diffusivity is of the conjectured order. We also obtain a pointwise upper bound for $D(t)$ the correct order.

math.PR↗

arXiv2007

An annihilating--branching particle model for the heat equation with average temperature zero

We consider two species of particles performing random walks in a domain in $\mathbb{R}^d$ with reflecting boundary conditions, which annihilate on contact. In addition, there is a conservation law so that the total number of particles of each type is preserved: When the two particles of different species annihilate each other, particles of each species, chosen at random, give birth. We assume initially equal numbers of each species and show that the system has a diffusive scaling limit in which the densities of the two species are well approximated by the positive and negative parts of the solution of the heat equation normalized to have constant $L^1$ norm. In particular, the higher Neumann eigenfunctions appear as asymptotically stable states at the diffusive time scale.

math.PR↗

arXiv2006

Bulk diffusion in a system with site disorder

We consider a system of random walks in a random environment interacting via exclusion. The model is reversible with respect to a family of disordered Bernoulli measures. Assuming some weak mixing conditions, it is shown that, under diffusive scaling, the system has a deterministic hydrodynamic limit which holds for almost every realization of the environment. The limit is a nonlinear diffusion equation with diffusion coefficient given by a variational formula. The model is nongradient and the method used is the ``long jump'' variation of the standard nongradient method, which is a type of renormalization. The proof is valid in all dimensions.

math.PR↗

arXiv2006

KdV Preserves White Noise

It is shown that white noise is an invariant measure for the Korteweg-deVries equation on $\mathbb T$. This is a consequence of recent results of Kappeler and Topalov establishing the well-posedness of the equation on appropriate negative Sobolev spaces, together with a result of Cambronero and McKean that white noise is the image under the Miura transform (Ricatti map) of the (weighted) Gibbs measure for the modified KdV equation, proven to be invariant for that equation by Bourgain.

math.AP↗

arXiv2006

$t^{1/3}$ Superdiffusivity of Finite-Range Asymmetric Exclusion Processes on $\mathbb Z$

We consider finite-range asymmetric exclusion processes on $\mathbb Z$ with non-zero drift. The diffusivity $D(t)$ is expected to be of ${\mathcal O}(t^{1/3})$. We prove that $D(t)\ge Ct^{1/3}$ in the weak (Tauberian) sense that $\int_0^\infty e^{-λt}tD(t)dt \ge Cλ^{-7/3}$ as $λ\to 0$. The proof employs the resolvent method to make a direct comparison with the totally asymmetric simple exclusion process, for which the result is a consequence of the scaling limit for the two-point function recently obtained by Ferrari and Spohn. In the nearest neighbor case, we show further that $tD(t)$ is monotone, and hence we can conclude that $D(t)\ge Ct^{1/3}(\log t)^{-7/3}$ in the usual sense.

math.PR↗

arXiv2006

Fluctuations of the front in a one dimensional model of X+Y-->2X

We consider a model of the reaction $X+Y\to 2X$ on the integer lattice in which $Y$ particles do not move while $X$ particles move as independent continuous time, simple symmetric random walks. $Y$ particles are transformed instantaneously to $X$ particles upon contact. We start with a fixed number $a\ge 1$ of $Y$ particles at each site to the right of the origin, and define a class of configurations of the $X$ particles to the left of the origin having a finite $l^1$ norm with a specified exponential weight. Starting from any configuration of $X$ particles to the left of the origin within such a class, we prove a central limit theorem for the position of the rightmost visited site of the $X$ particles.

math.PR↗

arXiv2005

Fluctuations of the front in a stochastic combustion model

We consider an interacting particle system on the one dimensional lattice $\bf Z$ modeling combustion. The process depends on two integer parameters $2\le a<M<\infty$. Particles move independently as continuous time simple symmetric random walks except that 1. When a particle jumps to a site which has not been previously visited by any particle, it branches into $a$ particles; 2. When a particle jumps to a site with $M$ particles, it is annihilated. We start from a configuration where all sites to the left of the origin have been previously visited and study the law of large numbers and central limit theorem for $r_t$, the rightmost visited site at time $t$. The proofs are based on the construction of a renewal structure leading to a definition of regeneration times for which good tail estimates can be performed.

math.PR↗

arXiv2003

Comparison inequality and two block estimate for inhomogeneous Bernoulli measures

We consider inhomogeneous Bernoulli measures of the form $\prod_{x\inΛ} p_x$ where $p_x$ are prescribed and uniformly bounded above and below away from 0 and 1. A comparison inequality is proved between the Kawasaki and Bernoulli-Laplace Dirichlet forms. Together with a recent result of Caputo on the gap of the Bernoulli-Laplace model, this proves a spectral gap of the correct order L^{-2} on cubes of side length L for the Kawasaki dynamics. The two block estimate of hydrodynamic limits is also obtained.

math.PR↗