Exact distributions of the maximum and range of random diffusivity processes

Sposini, Vittoria; Metzler, Ralf; Oshanin, Gleb; Seno, Flavio

doi:10.1088/1367-2630/abd313

Cited by 14 publications

(8 citation statements)

References 95 publications

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“…These works studied, amongst other things, Lévy walks, renewal processes, continuous time random walks and the Lévy Lorentz gas. Other than the BJP, the statistics of extreme events are studied for many different correlated physical systems [13][14][15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Big jump principle for heavy-tailed random walks with correlated increments

Höll,

Barkai

2021

Preprint

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The big jump principle explains the emergence of extreme events for physical quantities modelled by a sum of independent and identically distributed random variables which are heavy-tailed. Extreme events are large values of the sum and they are solely dominated by the largest summand called the big jump. Recently, the principle was introduced into physical sciences where systems usually exhibit correlations. Here, we study the principle for a random walk with correlated increments. Examples are the autoregressive model of first order and the discretized Ornstein-Uhlenbeck process both with heavy-tailed noise. The correlation leads to the dependence of large values of the sum not only on the big jump but also on the following increments. We describe this behaviour by two big jump principles, namely unconditioned and conditioned on the step number when the big jump occurs. The unconditional big jump principle is described by a correlation dependent shift between the sum and maximum distribution tails. For the conditional big jump principle, the shift depends also on the step number of the big jump.

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Section: Introductionmentioning

confidence: 99%

Big jump principle for heavy-tailed random walks with correlated increments

Höll,

Barkai

2021

Preprint

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“…Note that the span L(t) has the dimension of a length while V p (n) is dimensionless. The distribution of the span is a central quantity in the context of extreme value theory [67] and has been investigated for a wide range of stochastic processes [68,69,70,74,75,76,77]. Here, we show that for rt 1, the distribution P r (L, t) of the span L(t) of a BM with resetting rate r assumes the scaling form…”

Section: Summary Of the Main Resultsmentioning

confidence: 90%

Number of distinct sites visited by a resetting random walker

Biroli,

Mori,

Majumdar

2022

Preprint

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We investigate the number V p (n) of distinct sites visited by an nstep resetting random walker on a d-dimensional hypercubic lattice with resetting probability p. In the case p = 0, we recover the well-known result that the average number of distinct sites grows for large n asWe observe that the recurrence-transience transition at d = 2 for standard random walks (without resetting) disappears in the presence of resetting. In the limit p → 0, we compute the exact crossover scaling function between the two regimes. In the onedimensional case, we derive analytically the full distribution of V p (n) in the limit of large n. Moreover, for a one-dimensional random walker, we introduce a new observable, which we call imbalance, that measures how much the visited region is symmetric around the starting position. We analytically compute the full distribution of the imbalance both for p = 0 and for p > 0. Our theoretical results are verified by extensive numerical simulations.

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“…[11][12][13][14][15], to name a few. However, the EV statistics of correlated RVs still remains largely unexplored [16,17].…”

Section: Introductionmentioning

confidence: 99%

Discrete sampling of correlated random variables modifies the long-time behavior of their extreme value statistics

Zarfaty,

Barkai,

Kessler

2021

Preprint

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We consider the extreme value statistics of correlated random variables that arise from a Langevin equation. Recently, it was shown that the extreme values of the Ornstein-Uhlenbeck process follow a different distribution than those originating from its equilibrium measure, composed of independent and identically distributed Gaussian random variables. Here, we first focus on the discretely sampled Ornstein-Uhlenbeck process, which interpolates between these two limits. We show that in the limit of large times, its extreme values converge to those of the equilibrium distribution, instead of those of the continuously sampled process. This finding folds for any positive sampling interval, with an abrupt transition at zero. We then analyze the Langevin equation for any force that gives rise to a stable equilibrium distribution. For forces which asymptotically grow with the distance from the equilibrium point, the above conclusion continues to hold, and the extreme values for large times correspond to those of independent variables drawn from the equilibrium distribution. However, for forces which asymptotically decay to zero with the distance, the discretely sampled extreme value statistics at large times approach those of the continuously sampled process.

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Exact distributions of the maximum and range of random diffusivity processes

Abstract: We study the extremal properties of a stochastic process x t defined by the Langevin equation x ̇ … Show more

Cited by 14 publications

References 95 publications

Big jump principle for heavy-tailed random walks with correlated increments

Big jump principle for heavy-tailed random walks with correlated increments

Number of distinct sites visited by a resetting random walker

Discrete sampling of correlated random variables modifies the long-time behavior of their extreme value statistics

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