Averaged large deviations for random walk in a random environment

Abstract:We consider the quenched and the averaged (or annealed) large deviation rate functions I q and I a for space-time and (the usual) space-only RWRE on Z d . By Jensen's inequality, I a ≤ I q . In the space-time case, when d ≥ 3 + 1, I q and I a are known to be equal on an open set containing the typical velocity ξ o . When d = 1 + 1, we prove that I q and I a are equal only at ξ o . Similarly, when d = 2 + 1, we show that I a < I q on a punctured neighborhood of ξ o . In the space-only case, we provide a class of non-nestling walks on Z d with d = 2 or 3, and prove that I q and I a are not identically equal on any open set containing ξ o whenever the walk is in that class. This is very different from the known results for non-nestling walks on Z d with d ≥ 4.

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“…See the proofs of Lemmas 6 and 12 of [21]. In particular, the desired lower bound for the smallest eigenvalue of H a is evident from Eq.…”

Section: Reducing To a Fractional Moment Estimatementioning

confidence: 97%

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

Yilmaz

Zeitouni

2010

Commun. Math. Phys.

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“…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%

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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“…(ii) I a is strictly convex and analytic on an open set A a containing ξ o (see [13,23]); (c) if the walk is nestling, then N a is a line segment containing the origin that can extend in one or both directions (see [22]); it cannot extend in both directions when d = 2 (see [30]) or when d ≥ 5 (see [1]); (d) if the walk is nestling, but (T,û) is satisfied for someû ∈ S d−1 , then:…”

Section: 5mentioning

confidence: 99%

“…(i) the origin is an endpoint of N a (see [20]); (ii) I a is strictly convex and analytic on an open set A a (see [23]);…”

Section: 5mentioning

confidence: 99%

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Harmonic functions, h-transform and large deviations for random walks in random environments in dimensions four and higher

Yilmaz¹

2011

Ann. Probab.

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We consider large deviations for nearest-neighbor random walk in a uniformly elliptic i.i.d. environment on Z d . There exist variational formulae for the quenched and averaged rate functions Iq and Ia, obtained by Rosenbluth and Varadhan, respectively. Iq and Ia are not identically equal. However, when d ≥ 4 and the walk satisfies the so-called (T) condition of Sznitman, they have been previously shown to be equal on an open set Aeq .For every ξ ∈ Aeq , we prove the existence of a positive solution to a Laplace-like equation involving ξ and the original transition kernel of the walk. We then use this solution to define a new transition kernel via the h-transform technique of Doob. This new kernel corresponds to the unique minimizer of Varadhan's variational formula at ξ. It also corresponds to the unique minimizer of Rosenbluth's variational formula, provided that the latter is slightly modified.

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Averaged large deviations for random walk in a random environment

Cited by 14 publications

References 23 publications

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

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

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

Harmonic functions, h-transform and large deviations for random walks in random environments in dimensions four and higher

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