The cutoff phenomenon for random birth and death chains

Smith, Aaron

doi:10.1002/rsa.20693

Cited by 5 publications

(4 citation statements)

References 34 publications

(88 reference statements)

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“…This dynamical phase transition has been observed in a broad spectrum of random models. For instance, in the discrete state space the asymptotic cut-off phenomenon has been showed for shuffling cards Markov dynamics [1,32], random walks on the n-dimensional hypercube [56], birth and death Markov chains [19,74], sparse Markov chains [25], Glauber dynamics [33], SEP dynamics [40,52], SSEP dynamics [41], TASEP dynamics [34], random walks in random regular graphs [60], mean-zero field zero-range process [63], averaging processes [71], and sampling chains [7,51]. For more general Markov processes taking values in continuous state-spaces there are relatively few results showing the asymptotic cut-off phenomenon.…”

Section: Introductionmentioning

confidence: 99%

Cut-off phenomenon for Ornstein-Uhlenbeck processes driven by Lévy processes

Barrera¹,

Pardo²

2020

Electron. J. Probab.

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In this paper, we study the cut-off phenomenon of d-dimensional Ornstein-Uhlenbeck processes which are driven by Lévy processes under the total variation distance. To be more precise, we prove the abrupt convergence under the total variation distance of the aforementioned process to its equilibrium. Despite that the invariant distribution is not explicit, its distributional properties allow us to deduce that a profile function always exists in the reversible cases and it may exist in the non-reversible cases. The cut-off phenomenon for the average and superposition processes is also determined.

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Section: Introductionmentioning

confidence: 99%

Cut-off phenomenon for Ornstein-Uhlenbeck processes driven by Lévy processes

Barrera¹,

Pardo²

2020

Electron. J. Probab.

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“…Recently, part of the attention has shifted from "specific" to "generic" instances: instead of being fixed, the sequence of transition matrices itself is drawn at random from a certain distribution, and the cutoff phenomenon is shown to occur for almost every realization. Examples include certain random birth and death chains [14,23], "random random walks" on some finite groups [18,24], or the simple/non-backtracking random walk on various models of sparse random graphs, including random regular graphs [21], graphs with given degrees [7,6], and the giant component of the Erdös-Renyi random graph [7]. The above mentioned references are all concerned with the reversible case of undirected graphs, where the associated simple random walk and non-backtracking random walk have explicitly known stationary distributions.…”

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confidence: 99%

Cutoff at the “entropic time” for sparse Markov chains

Bordenave

Caputo

Salez

2018

Probab. Theory Relat. Fields

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We study convergence to equilibrium for a class of Markov chains in random environment. The chains are sparse in the sense that in every row of the transition matrix P the mass is essentially concentrated on few entries. Moreover, the entries are exchangeable within each row. This includes various models of random walks on sparse random directed graphs. The models are generally non reversible and the equilibrium distribution is itself unknown. In this general setting we establish the cutoff phenomenon for the total variation distance to equilibrium, with mixing time given by the logarithm of the number of states times the inverse of the average row entropy of P . As an application, we consider the case where the rows of P are i.i.d. random vectors in the domain of attraction of a Poisson-Dirichlet law with index α ∈ (0, 1). Our main results are based on a detailed analysis of the weight of the trajectory followed by the walker. This approach offers an interpretation of cutoff as an instance of the concentration of measure phenomenon.

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“…Another recent application of "the entropic method" for a problem related to repeated averages can be found in [12]. Lastly, we mention that cutoff was established also for random birth and death chains [15,29]. It would be interesting to establish the same for a natural model of a random walk on a random weighted tree.…”

Section: Related Work -Cutoff At the Entropic Time For Random Instanc...mentioning

confidence: 92%

Universality of cutoff for graphs with an added random matching

Hermon¹,

Sly²,

Sousi³

2020

Preprint

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We establish universality of cutoff for simple random walk on a class of random graphs defined as follows. Given a finite graph G = (V, E) with |V | even we define a random graph G * = (V, E ∪ E ) obtained by picking E to be the (unordered) pairs of a random perfect matching of V . We show that for a sequence of such graphs G n of diverging sizes and of uniformly bounded degree, if the minimal size of a connected component of G n is at least 3 for all n, then the random walk on G * n exhibits cutoff w.h.p. This provides a simple generic operation of adding some randomness to a given graph, which results in cutoff.

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The cutoff phenomenon for random birth and death chains

Cited by 5 publications

References 34 publications

Cut-off phenomenon for Ornstein-Uhlenbeck processes driven by Lévy processes

Cut-off phenomenon for Ornstein-Uhlenbeck processes driven by Lévy processes

Cutoff at the “entropic time” for sparse Markov chains

Universality of cutoff for graphs with an added random matching

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