Exact solution for a random walk in a time-dependent 1D random environment: the point-to-point Beta polymer

Thiery, Thimothé; Doussal, Pierre Le

doi:10.1088/1751-8121/50/4/045001

Cited by 20 publications

(40 citation statements)

References 54 publications

(256 reference statements)

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“…The second is extending the results at fixed u, to smaller values of u = x/t. From the results on the Beta polymer [1,11] we know that the PDF, P , exhibits additional local fluctuations, with a Gamma distribution, not present in the CDF, P > , and this distinction does not seem to be captured by the above arguments.…”

Section: First Model: Physical Argument and Length Scalesmentioning

confidence: 95%

“…For the continuum diffusion model II, using J(u) and λ(u) given in (11), the arguments in Section 2 predict, in the limit of small γ, precisely in an expansion in…”

Section: Large Deviation Regime: Tw Fluctuationsmentioning

confidence: 95%

“…where χ GUE denotes a random variable following the Tracy-Widom GUE distribution, for some explicit functions I(x), σ(x) [1,11] which depend on α, β.…”

Section: Convergence To the Kpz Equationmentioning

confidence: 99%

“…Note that the exact solvability of the Beta RWRE is rooted in a work of Povolotsky [10] on Bethe ansatz solvable probabilistic models of interacting particles. The connection between the Beta RWRE and the KPZ universality class was further strengthened and refined in [11,12].…”

Section: Introductionmentioning

confidence: 99%

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Moderate deviations for diffusion in time dependent random media

Barraquand¹,

Doussal²

2020

J. Phys. A: Math. Theor.

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The position x(t) of a particle diffusing in a one-dimensional uncorrelated and time dependent random medium is simply Gaussian distributed in the typical direction, i.e. along the ray x = v0t, where v0 is the average drift. However, it has been found that it exhibits at large time sample to sample fluctuations characteristic of the KPZ universality class when observed in an atypical direction, i.e. along the ray x = vt with v = v0. Here we show, from exact solutions, that in the moderate deviation regime x − v0t ∝ t 3/4 these fluctuations are precisely described by the finite time KPZ equation, which thus describes the crossover between the Gaussian typical regime and the KPZ fixed point regime for the large deviations. This confirms heuristic arguments given in [2]. These exact results include the discrete model known as the Beta RWRE, and a continuum diffusion. They predict the behavior of the maximum of a large number of independent walkers, which should be easier to observe (e.g. in experiments) in this moderate deviations regime. arXiv:1912.11085v1 [cond-mat.stat-mech]

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Section: First Model: Physical Argument and Length Scalesmentioning

confidence: 95%

“…For the continuum diffusion model II, using J(u) and λ(u) given in (11), the arguments in Section 2 predict, in the limit of small γ, precisely in an expansion in…”

Section: Large Deviation Regime: Tw Fluctuationsmentioning

confidence: 95%

“…where χ GUE denotes a random variable following the Tracy-Widom GUE distribution, for some explicit functions I(x), σ(x) [1,11] which depend on α, β.…”

Section: Convergence To the Kpz Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Moderate deviations for diffusion in time dependent random media

Barraquand¹,

Doussal²

2020

J. Phys. A: Math. Theor.

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“…Recently, Barraquand and Corwin obtained an exact solution of a discrete TD-RWRE on Z with SR correlated jump probabilities, the so-called Beta polymer. The sample to sample fluctuations of the logarithm of the cumulative [31] and transition [32] probability distribution function (PDF) in the large deviations regime of the RW, ie. looking away from the most probable direction, were found to be distributed with the characteristic KPZ exponent and GUE TW distribution 1 .…”

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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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Tracy-Widom Asymptotics for a River Delta Model

Barraquand¹,

Rychnovsky²

2019

Springer Proceedings in Mathematics &Amp; Statistics

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We study an oriented first passage percolation model for the evolution of a river delta. This model is exactly solvable and occurs as the low temperature limit of the beta random walk in random environment. We analyze the asymptotics of an exact formula from [13] to show that, at any fixed positive time, the width of a river delta of length L approaches a constant times L 2/3 with Tracy-Widom GUE fluctuations of order L 4/9 . This result can be rephrased in terms of particle systems. We introduce an exactly solvable particle system on the integer half line and show that after running the system for only finite time the particle positions have Tracy-Widom fluctuations.

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Exact solution for a random walk in a time-dependent 1D random environment: the point-to-point Beta polymer

Cited by 20 publications

References 54 publications

Moderate deviations for diffusion in time dependent random media

Moderate deviations for diffusion in time dependent random media

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

Tracy-Widom Asymptotics for a River Delta Model

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