Functional weak limit of random walks in cooling random environment

Xie, Yongjia

doi:10.1214/20-ecp360

Cited by 2 publications

(1 citation statement)

References 14 publications

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“…State of the art of RWCRE. The study of RWCRE in one-dimension has been pursued in a sequence of recent works [6,3,4,5,30] in various regimes which we next briefly describe.…”

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confidence: 99%

Gaussian, stable, tempered stable and mixed limit laws for random walks in cooling random environments

Avena¹,

Costa²,

Peterson³

2021

Preprint

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Random Walks in Cooling Random Environments (RWCRE) is a model of random walks in dynamic random environments where the entire environment is resampled along a fixed sequence of times, called the "cooling sequence", and is kept fixed in between those times. This model interpolates between that of a homogenous random walk, where the environment is reset at every step, and Random Walks in (static) Random Environments (RWRE), where the environment is never resampled. In this work we focus on the limiting distributions of one-dimensional RWCRE in the regime where the fluctuations of the corresponding (static) RWRE is given by a s-stable random variable with s ∈ (1, 2). In this regime, due to the two extreme cases (resampling every step and never resampling, respectively), a crossover from Gaussian to stable limits for sufficiently regular cooling sequence was previously conjectured. Our first result answers affirmatively this conjecture by making clear critical exponent, norming sequences and limiting laws associated with the crossover which demonstrates a change from Gaussian to s-stable limits, passing at criticality through a certain generalized tempered stable distribution. We then explore the resulting RWCRE scaling limits for general cooling sequences. On the one hand, we offer sets of operative sufficient conditions that guarantee asymptotic emergence of either Gaussian, s-stable or generalized tempered distributions from a certain class. On the other hand, we give explicit examples and describe how to construct irregular cooling sequences for which the corresponding limit law is characterized by mixtures of the three above mentioned laws. To obtain these results, we need and derive a number of refined asymptotic results for the static RWRE with s ∈ (1, 2) which may be of independent interest.

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“…State of the art of RWCRE. The study of RWCRE in one-dimension has been pursued in a sequence of recent works [6,3,4,5,30] in various regimes which we next briefly describe.…”

mentioning

confidence: 99%

Gaussian, stable, tempered stable and mixed limit laws for random walks in cooling random environments

Avena¹,

Costa²,

Peterson³

2021

Preprint

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Branching processes in random environment with cooling

Korshunov

2023

Дискрет. матем.

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Известно, что ветвящийся процесс в случайной среде тесно связан с соответствующим случайным блужданием $S_n = \xi_1 + \dotsb + \xi_n$, где $\xi_k = \ln \varphi_{\eta_k}'(1)$. Здесь $\varphi_x (t)$ и $\{ \eta_k \}_{k = 1}^{\infty}$ - производящая функция числа потомков и случайная среда соответственно. В статье изучается вероятность вырождения ветвящегося процесса в случайной среде с замораживаниями при $\mathsf{E} \xi_1 > 0$, отличающегося от обычного ветвящегося процесса в случайной среде тем, что каждое значение среды устанавливается на несколько поколений. Оказывается, что такой процесс также тесно связан со случайным блужданием $S_n = \tau_1 \xi_1 + \dotsb + \tau_n \xi_n$, где $\xi_k = \ln \varphi_{\eta_k}'(1)$. Здесь $\varphi_x (t)$ и $\{ \eta_k \}_{k = 1}^{\infty}$ - производящая функция числа потомков одной частицы при условии среды $x$ и случайная среда соответственно, а $\tau_k$ - длительность $k$-го замораживания. Статья содержит несколько условий, достаточных для невырождения процесса с положительной вероятностью, и несколько условий, достаточных для вырождения процесса с вероятностью $1$.

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Functional weak limit of random walks in cooling random environment

Cited by 2 publications

References 14 publications

Gaussian, stable, tempered stable and mixed limit laws for random walks in cooling random environments

Gaussian, stable, tempered stable and mixed limit laws for random walks in cooling random environments

Branching processes in random environment with cooling

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