Differing Averaged and Quenched Large Deviations for Random Walks in Random Environments in Dimensions Two and Three

Yilmaz, Atilla; Zeitouni, Ofer

doi:10.1007/s00220-010-1119-3

Cited by 31 publications

(38 citation statements)

References 26 publications

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“…This is indeed in agreement with (39) and (42). Indeed the term of order O(α) 2 corresponds to the rescaling involving (39), the term of order O(α) corresponds to the rescaling involving…”

Section: Supplemental Materialssupporting

confidence: 81%

“…This picture agrees with known results. It was indeed shown in [37][38][39] that the annealed and quenched large deviations rate functions of an unbiased lattice RW, respectively defined as I a (| u|) := − lim t→∞ ln Q(t u,t) t , and…”

Section: Main Analysismentioning

confidence: 99%

“…Let us now compare (43)-(44) and (42) using the scaling limit (39). Using the weak noise limit of the Beta polymer, the result (43)-(44) should imply (42).…”

Section: Supplemental Materialsmentioning

confidence: 99%

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Diffusion in time-dependent random media and the Kardar-Parisi-Zhang equation

Doussal

Thiery

2017

Phys. Rev. E

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Although time-dependent random media with short range correlations lead to (possibly biased) normal tracer diffusion, anomalous fluctuations occur away from the most probable direction. This was pointed out recently in 1D lattice random walks, where statistics related to the 1D KardarParisi-Zhang (KPZ) universality class, i.e. the GUE Tracy Widom distribution, were shown to arise. Here we provide a simple picture for this correspondence, directly in the continuum as well as for lattice models, which allows to study arbitrary space dimension and to predict a variety of universal distributions. In d = 1 we predict and verify numerically the emergence of the GOE Tracy-Widom distribution for the fluctuations of the transition probability. In d = 3 we predict a phase transition from Gaussian fluctuations to 3D-KPZ type fluctuations as the bias is increased. We predict KPZ universal distributions for the arrival time of a first particle from a cloud diffusing in such media.Introduction Diffusion in random media arises in numerous fields, e.g. oil exploration in porous rocks [1], spreading of pollutants in inhomogeneous flows [2], diffusion of charge carriers in conductors [3], relaxation properties of glasses [4], defect motions in solids, econophysics, population dynamics [5,6]. Many works have studied time independent, i.e. static, random environments [7], in d = 1 [8] or in higher dimensions, with short-range (SR) [9,10] of long-range (LR) spatial correlations [11]. It was found that static disorder with SR correlations is generically irrrelevant above the upper-critical dimension While particles tyically diffuse normally as if the random environment had been averaged out, particles conditioned on arriving away from the Gaussian bulk of the distribution in the 'large large deviations regime' (for |x(t)| > uct) are superdiffusive with the roughness exponent of the directed polymer in the pinned phase ζ d > 1/2. In this regime, fluctuations of the logarithm of the transition probability are large (scale with t θ d with θ d = −1 + 2ζ d > 0) and identical to those of the height in the rough phase of the KPZ equation. The two phases are separated by an Edward-Wilkinson regime of fluctuations when x = uct + o(t). In d = 1, 2 uc = 0 while for d ≥ 3 there is a phase transition with uc = 0. The picture is drawn here in the absence of a systematic bias f . In the presence of a bias the bulk is around x ∼ f t and the transition occurs for x = ut with |( f + u)| = uc.d c = 2, leading to normal diffusion in d = 3, while LR disorder can lead to anomalous diffusion in any d.Another important class of random media are timedependent. These have been studied e.g. in the context of wave propagation [12], dispersion of particles in turbulent flows [2] (the famous Richardson's law [13]), and in the problem of the passive scalar [14]. The latter cases involve long range correlations in the flow, and lead to anomalous transport or multiscaling. The, a priori more benign, case of SR space-time correlations has received much attention rece...

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Section: Supplemental Materialssupporting

confidence: 81%

Section: Main Analysismentioning

confidence: 99%

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Diffusion in time-dependent random media and the Kardar-Parisi-Zhang equation

Doussal

Thiery

2017

Phys. Rev. E

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“…These results were later generalized by Rassoul-Agha et al [25][26][27][28]. Moreover, Peterson et al [22,30,31] recently studied some properties of the quenched and the annealed rate functions. However, we still have less information about these rate functions than is known in the onedimensional case.…”

Section: Introductionmentioning

confidence: 85%

Continuity for the Rate Function of the Simple Random Walk on Supercritical Percolation Clusters

Kubota

2019

J Theor Probab

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We consider the simple random walk on supercritical percolation clusters in the multidimensional cubic lattice. In this model, a quenched large deviation principle holds for the position of the random walk. Its rate function depends on the law of the percolation configuration, and the aim of this paper is to study the continuity of the rate function in the law. To do this, it is useful that the rate function is expressed by the so-called Lyapunov exponent, which is the asymptotic cost paid by the random walk for traveling in a landscape of percolation configurations. In this context, we first observe the continuity of the Lyapunov exponent in the law of the percolation configuration, and then lift it to the rate function.

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“…Some of these results, which include level-3 LDP can be found in [27], [41], [42], [28], [30], [29]. Yilmaz and Zeitouni [43] studied a class of random walks in a random environment where the annealed and quenched rate functions differ.…”

Section: Introductionmentioning

confidence: 99%

Large deviations for Brownian motion in a random potential

Boivin

Lê

2020

ESAIM: PS

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A quenched large deviation principle for Brownian motion in a non-negative, stationary potential is proved. A sufficient moment condition on the potential is given but unlike the results of Armstrong and Tran (2014) no regularity is assumed. The proof is based on a method developed by Sznitman (1994) for Brownian motion among Poissonian potential. In particular, the LDP holds for potentials with polynomially decaying correlations such as the classical potentials studied by L. Pastur (1977) and R. Fukushima (2008) and the potentials recently introduced by H. Lacoin (2012).Subject Classification: 82B41, 60K37

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Differing Averaged and Quenched Large Deviations for Random Walks in Random Environments in Dimensions Two and Three

Cited by 31 publications

References 26 publications

Diffusion in time-dependent random media and the Kardar-Parisi-Zhang equation

Diffusion in time-dependent random media and the Kardar-Parisi-Zhang equation

Continuity for the Rate Function of the Simple Random Walk on Supercritical Percolation Clusters

Large deviations for Brownian motion in a random potential

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