arXiv subjects

Enrique Guerra

Publications and source records attributed to Enrique Guerra.

4 recordsLinked to original sources

arXiv2021

Polynomial ballisticity conditions in mixing environments

We prove ballistic behaviour as well as an annealed functional central limit theorem for random walks in mixing random environments (RWRE). The ballistic hypothesis will be an effective polynomial condition as the one introduced by Berger, Drewitz, and Ram\'ırez (\emph{Comm. Pure Appl. Math,} {\bf 67}, (2014) 1947--1973). The novel idea therein was the construction of several simultaneous renormalization steps, providing more flexibility for seed estimates. For our proof, we indeed follow a similar path, and introduce a new mixing effective criterion which will be implied by the polynomial condition. This allows us to prove, in a mixing framework, the RWRE conjecture concerning the equivalence between each condition $(T^γ)|\ell$, for $γ\in (0,1)$ and $\ell \in \mathbb S^{d-1}$. This work complements the previous work of Guerra (\emph{Ann. Probab.} {\bf 47} (2019) 3003--3054) and completes the answer about the meaning of condition $(T')|\ell$ in a mixing setting, an open question posed by Comets and Zeitouni (\emph{Ann. Probab.} {\bf 32} (2004) 880--914).

math.PR↗

arXiv2020

On the connection between transient and ballistic behaviours for RWRE

We study the strong form of the ballistic conjecture for random walks in random environments (RWRE). This conjecture asserts that any RWRE which is directionally transient for a nonempty open set of directions satisfies condition $(T)$ (annealed exponential decay for the unlikely exit probability). Specifically, we introduce a ballisticity condition which is fulfilled as soon as a polynomial condition of degree greater than $d-1$ holds. Under that hypothesis we prove condition $(T)$, which turns this condition into the weakest-known ballisticity assumption. We recall that standard arguments to prove that a ballisticity condition implies directional transience require at least polynomial decay greater than degree $d$. Furthermore, in the one dimensional case we provide an alternative proof which proves the equivalence between transient behaviour and annealed arbitrary decay for the unlikely exit probability, we expect that this new argument might be used in higher dimensions.

math.PR↗

arXiv2018

A proof of Sznitman's conjecture about ballistic RWRE

We consider a random walk in a uniformly elliptic i.i.d. random environment in $\mathbb Z^d$ for $d\ge 2$. It is believed that whenever the random walk is transient in a given direction it is necessarily ballistic. In order to quantify the gap which would be needed to prove this equivalence, several ballisticity conditions have been introduced. In particular, in 2001 and 2002, Sznitman defined the so called conditions $(T)$ and $(T')$. The first one is the requirement that certain unlikely exit probabilities from a set of slabs decay exponentially fast with their width $L$. The second one is the requirement that for all $γ\in (0,1)$ condition $(T)_γ$ is satisfied, which in turn is defined as the requirement that the decay is like $e^{-CL^γ}$ for some $C>0$. In this article we prove a conjecture of Sznitman of 2002, stating that $(T)$ and $(T')$ are equivalent. Hence, this closes the circle proving the equivalence of conditions $(T)$, $(T')$ and $(T)_γ$ for some $γ\in (0,1)$ as conjectured by Sznitman, and also of each of these ballisticity conditions with the polynomial condition $(P)_M$ for $M\ge 15d+5$ introduced by Berger, Drewitz and Ramirez in 2014.

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↗