Browsing by Author "Ramirez, Alejandro F."
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- ItemA non-oriented first passage percolation model and statistical invariance by time reversal(2024) Ramirez, Alejandro F.; Saglietti, Santiago; Shao, LingyunWe introduce and study a non -oriented first passage percolation model having a property of statistical invariance by time reversal. This model is defined in a graph having directed edges and the passage times associated with each set of outgoing edges from a given vertex are distributed according to a generalized Bernoulli-Exponential law and i.i.d. among vertices. We derive the statistical invariance property by time reversal through a zero -temperature limit of the random walk in Dirichlet environment model.
- ItemAsymptotic direction in random walks in random environment revisited(BRAZILIAN STATISTICAL ASSOCIATION, 2010) Drewitz, Alexander; Ramirez, Alejandro F.Consider a random walk {X(n) : n >= 0} in an elliptic i.i.d. environment in dimensions d >= 2 and call P(0) its averaged law starting from 0. Given a direction I is an element of S(d-1), A(l) = {lim(n ->infinity) Xn . l = infinity} is called the event that the random walk is transient in the direction I. Recently Simenhaus proved that the following are equivalent: the random walk is transient in the neighborhood of a given direction; P(0)-a.s. there exists a deterministic asymptotic direction; the random walk is transient in any direction contained in the open half space defined by this asymptotic direction. Here we prove that the following are equivalent: P(0)(A(l) boolean OR A(-l)) = 1 in the neighborhood of a given direction; there exists an asymptotic direction v such that P(0) (A(upsilon) boolean OR A(-upsilon)) = 1 and P(0)-a.s we have lim(n ->infinity) X(n)/vertical bar X(n)vertical bar = 1(A upsilon)upsilon - 1(A-upsilon)upsilon; P(0) (A(l) boolean OR A(-l)) = 1 if and only if l . upsilon not equal 0. Furthermore, we give a review of some open problems.
- ItemFLUCTUATIONS OF THE FRONT IN A ONE DIMENSIONAL MODEL OF X plus Y -> 2X(AMER MATHEMATICAL SOC, 2009) Comets, Francis; Quastel, Jeremy; Ramirez, Alejandro F.We consider a model of the reaction X + Y -> 2X on the integer lattice in which Y particles do not move while X particles move its independent continuous time, simple symmetric random walks. Y particles are transformed instantaneously to X particles upon contact We start with a fixed number a >= 1 of Y particles at each site to the right of the origin We prove a central limit theorem for the rightmost visited site of the X particles lip to time t and show that the of the environment as seen from the front converges to a unique invariant measure
- ItemFluctuations of the front in a stochastic combustion model(GAUTHIER-VILLARS/EDITIONS ELSEVIER, 2007) Comets, Francis; Quastel, Jeremy; Ramirez, Alejandro F.We consider an interacting particle system on the one-dimensional lattice Z modeling combustion. The process depends on two integer parameters 2 <= a <= M <= infinity. Particles move independently as continuous time simple symmetric random walks except that (i) when a particle jumps to a site which has not been previously visited by any particle, it branches into a particles, (ii) 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. (c) 2006 Elsevier Masson SAS. All rights reserved.
- ItemON A GENERAL MANY-DIMENSIONAL EXCITED RANDOM WALK(INST MATHEMATICAL STATISTICS, 2012) Menshikov, Mikhail; Popov, Serguei; Ramirez, Alejandro F.; Vachkovskaia, MarinaIn this paper we study a substantial generalization of the model of excited random walk introduced in [Electron. Commun. Probab. 8 (2003) 86-92] by Benjamini and Wilson. We consider a discrete-time stochastic process (X-n, n = 0, 1, 2, ...) taking values on Z(d), d >= 2, described as follows: when the particle visits a site for the first time, it has a uniformly-positive drift in a given direction l; when the particle is at a site which was already visited before, it has zero drift. Assuming uniform ellipticity and that the jumps of the process are uniformly bounded, we prove that the process is ballistic in the direction l so that lim inf(n ->infinity) X-n.l/n > 0. A key ingredient in the proof of this result is an estimate on the probability that the process visits less than n(1/2+alpha) distinct sites by time n, where a is some positive number depending on the parameters of the model. This approach completely avoids the use of tan points and coupling methods specific to the excited random walk. Furthermore, we apply this technique to prove that the excited random walk in an i.i.d. random environment satisfies a ballistic law of large numbers and a central limit theorem.
- ItemQUENCHED AND AVERAGED LARGE DEVIATIONS FOR RANDOM WALKS IN RANDOM ENVIRONMENTS: THE IMPACT OF DISORDER(2023) Bazaes, Rodrigo; Mukherjee, Chiranjib; Ramirez, Alejandro F.; Saglietti, SantiagoIn 2003, Varadhan (Comm. Pure Appl. Math. 56 (2003) 1222-1245) de-veloped a robust method for proving quenched and averaged large deviations for random walks in a uniformly elliptic and i.i.d. environment (RWRE) on Zd. One fundamental question which remained open was to determine when the quenched and averaged large deviation rate functions agree, and when they do not. In this article we show that for RWRE in uniformly elliptic and i.i.d. environment in d > 4, the two rate functions agree on any compact set contained in the interior of their domain which does not contain the origin, provided that the disorder of the environment is sufficiently low. Our result provides a new formulation which encompasses a set of sufficient conditions under which these rate functions agree without assuming that the RWRE is ballistic (see (Probab. Theory Related Fields 149 (2011) 463-491)), satis-fies a CLT or even a law of large numbers (Electron. Commun. Probab. 7 (2002)191-197; Ann. Probab. 36 (2008) 728-738). Also, the equality of rate functions is not restricted to neighborhoods around given points, as long as the disorder of the environment is kept low. One of the novelties of our ap-proach is the introduction of an auxiliary random walk in a deterministic envi-ronment which is itself ballistic (regardless of the actual RWRE behavior) and whose large deviation properties approximate those of the original RWRE in a robust manner, even if the original RWRE is not ballistic itself.
- ItemSecond order cubic corrections of large deviations for perturbed random walks*(2022) Oviedo, Giancarlos; Panizo, Gonzalo; Ramirez, Alejandro F.We prove that the Beta random walk, introduced in [BC17] 2017, has cubic fluctuations from the large deviation principle of the GUE Tracy-Widom type for arbitrary values ?? > 0 and (3 > 0 of the parameters of the Beta distribution, removing previous restrictions on their values. Furthermore, we prove that the GUE Tracy-Widom fluctuations still hold in the intermediate disorder regime. We also show that any random walk in space-time random environment that matches certain moments with the Beta random walk also has GUE Tracy-Widom fluctuations in the intermediate disorder regime. As a corollary we show the emergence of GUE Tracy-Widom fluctuations from the large deviation principle for trajectories ending at boundary points for random walks in space (time-independent) i.i.d. Dirichlet random environment in dimension d = 2 for a class of asymptotic behavior of the parameters.
- ItemThe effect of disorder on quenched and averaged large deviations for random walks in random environments: Boundary behavior(2023) Bazaes, Rodrigo; Mukherjee, Chiranjib; Ramirez, Alejandro F.; Saglietti, SantiagoFor a random walk in a uniformly elliptic and i.i.d. environment on Zd with d >= 4, we show that the quenched and annealed large deviation rate functions agree on any compact set contained in the boundary an := {x is an element of Rd : |x|1 = 1} of their domain which does not intersect any of the (d - 2)-dimensional facets of an, provided that the disorder of the environment is low enough (depending on the compact set). As a consequence, we obtain a simple explicit formula for both rate functions on any such compact set of an at low enough disorder. In contrast to previous works, our results do not assume any ballistic behavior of the random walk and are not restricted to neighborhoods of any given point (on the boundary an). In addition, our results complement those in Bazaes et al. (2022), where, using different methods, we investigate the equality of the rate functions in the interior of their domain. Finally, for a general parametrized family of environments, we show that the strength of disorder determines a phase transition in the equality of both rate functions, in the sense that for each x is an element of an there exists ex such that the two rate functions agree at x when the disorder is smaller than ex and disagree when it is larger. This further reconfirms the idea, introduced in Bazaes et al. (2022), that the disorder of the environment is in general intimately related with the equality of the rate functions.(c) 2023 Elsevier B.V. All rights reserved.