Searcharxiv⌕ Search

arXiv subjects

Alejandro F. Ramirez

Publications and source records attributed to Alejandro F. Ramirez.

12 recordsLinked to original sources
arXiv2018

Random polymers on the complete graph

Consider directed polymers in a random environment on the complete graph of size $N$. This model can be formulated as a product of i.i.d. $N\times N$ random matrices and its large time asymptotics is captured by Lyapunov exponents and the Furstenberg measure. We detail this correspondence, derive the long-time limit of the model and obtain a co-variant distribution for the polymer path. Next, we observe that the model becomes exactly solvable when the disorder variables are located on edges of the complete graph and follow a totally asymmetric stable law of index $α\in (0,1)$. Then, a certain notion of mean height of the polymer behaves like a random walk and we show that the height function is distributed around this mean according to an explicit law. Large $N$ asymptotics can be taken in this setting, for instance, for the free energy of the system and for the invariant law of the polymer height with a shift. Moreover, we give some perturbative results for environments which are close to the totally asymmetric stable laws.

math.PR↗
arXiv2017

Random walk in the low disorder ballistic regime

We consider a random walk in $\mathbb Z^d$ which jumps from a site $x$ to a nearest neighboring site $x+e$ (where $e\in V:=\{x\in\mathbb Z^d: |x|_1=1\}$) with probability $p_0(e)+εξ(x,e)$. Here $\sum_e p_0(e)=1$, $p_0(e)> 0$, $ε$ is a small parameter while $\{\{ξ(x,e):e\in V\}: x\in\mathbb Z^d\}$ are i.i.d. random variables with an absolute value bounded by $1$. We review recent progress in the non-vanishing velocity case, giving an asymptotic expansion in $ε$ of the invariant measure of the environmental process, and bounds for the velocity.

math.PR↗
arXiv2016

Exponential ergodicity and Rayleigh-Schroedinger series for infinite dimensional diffusions

We consider an infinite dimensional diffusion on $T^{\mathbb Z^d}$, where $T$ is the circle, defined by an infinitesimal generator of the form $L=\sum_{i\in\mathbb Z^d}\left(\frac{a_i(η)}{2}\partial^2_i +b_i(η)\partial_i\right)$, with $η\in T^{\mathbb Z^d}$, where the coefficients $a_i,b_i$ are of finite range, bounded with uniformly bounded second order partial derivatives and the ellipticity assumption $\inf_{i,η}a_i(η)>0$ is satisfied. We prove that whenever $ν$ is an invariant Gibbs measure for this diffusion satisfying the logarithmic Sobolev inequality, then the dynamics is exponentially ergodic in the uniform norm, and hence $ν$ is the unique invariant measure. As an application of this result, we prove that if $A=\sum_{i\in\mathbb Z^d}c_i(η)\partial_i$, and $c_i$ satisfy the condition $\sum_{i\in\mathbb Z^d} \int c_i^2dν<\infty$, then there is an $ε_c>0$, such that for every $ε\in (-ε_c,ε_c)$, the infinite dimensional diffusion with generator $L_ε=L+εA$, has a unique invariant measure $ν_ε$ having a Radon-Nikodym derivative $g_ε$ with respect to $ν$, which admits the analytic expansion $g_ε=\sum_{k=0}^\infty ε^k f_k$, where $f_k\in L_2[ν]$ are defined through $f_0=1$, $\int f_kdν=0$ and the recurrence equations $L^*f_{k+1}=A^*f_k$. We give an example where through this expansion we are able to quantify the effect on the invariant measure of a perturbation triggering interaction on independent diffusions.

math.PR↗
arXiv2015

Asymptotic expansion of the invariant measure for ballistic random walk in the low disorder regime

We consider a random walk in random environment in the low disorder regime on $\mathbb Z^d$. That is, the probability that the random walk jumps from a site $x$ to a nearest neighboring site $x+e$ is given by $p(e)+εξ(x,e)$, where $p(e)$ is deterministic, $\{\{ξ(x,e):|e|_1=1\}:x\in\mathbb Z^d\}$ are i.i.d. and $ε>0$ is a parameter which is eventually chosen small enough. We establish an asymptotic expansion in $ε$ for the invariant measure of the environmental process whenever a ballisticity condition is satisfied. As an application of our expansion, we derive a numerical expression up to first order in $ε$ for the invariant measure of random perturbations of the simple symmetric random walk in dimensions $d=2$.

math.PR↗
arXiv2014

Almost exponential decay for the exit probability from slabs of ballistic RWRE

It is conjectured that in dimensions $d\ge 2$ any random walk in an i.i.d. uniformly elliptic random environment (RWRE) which is directionally transient is ballistic. The ballisticity conditions for RWRE somehow interpolate between directional transience and ballisticity and have served to quantify the gap which would need to be proven in order to answer affirmatively this conjecture. Two important ballisticity conditions introduced by Sznitman \cite{Sz02} in 2001 and 2002 are the so called conditions $(T')$ and $(T)$: given a slab of width $L$ orthogonal to $l$, condition $(T')$ in direction $l$ is the requirement that the annealed exit probability of the walk through the side of the slab in the half-space $\{x:x\cdot l<0\}$, decays faster than $e^{-CL^γ}$ for all $γ\in (0,1)$ and some constant $C>0$, while condition $(T)$ in direction $l$ is the requirement that the decay is exponential $e^{-CL}$. It is believed that $(T')$ implies $(T)$. In this article we show that $(T')$ implies at least an {\it almost} (in a sense to be made precise) exponential decay.

math.PR↗
arXiv2014

Ellipticity criteria for ballistic behavior of random walks in random environment

We introduce ellipticity criteria for random walks in i.i.d. random environments under which we can extend the ballisticity conditions of Sznitman's and the polynomial effective criteria of Berger, Drewitz and Ramirez originally defined for uniformly elliptic random walks. We prove under them the equivalence of Sznitman's (T') condition with the polynomial effective criterion (P)_M, for M large enough. We furthermore give ellipticity criteria under which a random walk satisfying the polynomial effective criterion, is ballistic, satisfies the annealed central limit theorem or the quenched central limit theorem.

math.PR↗
arXiv2013

Last passage percolation and traveling fronts

We consider a system of N particles with a stochastic dynamics introduced by Brunet and Derrida. The particles can be interpreted as last passage times in directed percolation on {1,...,N} of mean-field type. The particles remain grouped and move like a traveling wave, subject to discretization and driven by a random noise. As N increases, we obtain estimates for the speed of the front and its profile, for different laws of the driving noise. The Gumbel distribution plays a central role for the particle jumps, and we show that the scaling limit is a Lévy process in this case. The case of bounded jumps yields a completely different behavior.

math.PR↗
arXiv2012

Level 1 quenched large deviation principle for random walk in dynamic random environment

Consider a random walk in a time-dependent random environment on the lattice Zd. Recently, Rassoul-Agha, Seppalainen and Yilmaz [RSY11] proved a general large deviation principle under mild ergodicity assumptions on the random environment for such a random walk, establishing first level 2 and 3 large deviation principles. Here we present two alternative short proofs of the level 1 large deviations under mild ergodicity assumptions on the environment: one for the continuous time case and another one for the discrete time case. Both proofs provide the existence, continuity and convexity of the rate function. Our methods are based on the use of the sub-additive ergodic theorem as presented by Varadhan in 2003.

math.PR↗
arXiv2007

Transition asymptotics for reaction-diffusion in random media

We describe a universal transition mechanism characterizing the passage to an annealed behavior and to a regime where the fluctuations about this behavior are Gaussian, for the long time asymptotics of the empirical average of the expected value of the number of random walks which branch and annihilate on ${\mathbb Z}^d$, with stationary random rates. The random walks are independent, continuous time rate $2dκ$, simple, symmetric, with $κ\ge 0$. A random walk at $x\in{\mathbb Z}^d$, binary branches at rate $v_+(x)$, and annihilates at rate $v_-(x)$. The random environment $w$ has coordinates $w(x)=(v_-(x),v_+(x))$ which are i.i.d. We identify a natural way to describe the annealed-Gaussian transition mechanism under mild conditions on the rates. Indeed, we introduce the exponents $F_θ(t):=\frac{H_1((1+θ)t)-(1+θ)H_1(t)}θ$, and assume that $\frac{F_{2θ}(t)-F_θ(t)}{θ\log(κt+e)}\to\infty$ for $|θ|>0$ small enough, where $H_1(t):=\log < m(0,t)>$ and $ $ denotes the average of the expected value of the number of particles $m(0,t,w)$ at time $t$ and an environment of rates $w$, given that initially there was only one particle at 0. Then the empirical average of $m(x,t,w)$ over a box of side $L(t)$ has different behaviors: if $ L(t)\ge e^{\frac{1}{d} F_ε(t)}$ for some $ε>0$ and large enough $t$, a law of large numbers is satisfied; if $ L(t)\ge e^{\frac{1}{d} F_ε(2t)}$ for some $ε>0$ and large enough $t$, a CLT is satisfied. These statements are violated if the reversed inequalities are satisfied for some negative $ε$. Applications to potentials with Weibull, Frechet and double exponential tails are given.

math.PR↗
arXiv2007

Front propagation in an exclusion one-dimensional reactive dynamics

We consider an exclusion process representing a reactive dynamics of a pulled front on the integer lattice, describing the dynamics of first class $X$ particles moving as a simple symmetric exclusion process, and static second class $Y$ particles. When an $X$ particle jumps to a site with a $Y$ particle, their position is intechanged and the $Y$ particle becomes an $X$ one. Initially, there is an arbitrary configuration of $X$ particles at sites $..., -1,0$, and $Y$ particles only at sites $1,2,...$, with a product Bernoulli law of parameter $ρ,0<ρ<1$. We prove a law of large numbers and a central limit theorem for the front defined by the right-most visited site of the $X$ particles at time $t$. These results corroborate Monte-Carlo simulations performed in a similar context. We also prove that the law of the $X$ particles as seen from the front converges to a unique invariant measure. The proofs use regeneration times: we present a direct way to define them within this context.

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↗