Sampling biased monotonic surfaces using exponential metrics

Greenberg, Sam; Randall, Dana; Streib, Amanda Pascoe

doi:10.1017/s0963548320000188

Cited by 6 publications

(19 citation statements)

References 22 publications

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“…Our proof is similar to that of [10] and we defer it to Appendix A.3. The idea is that the hitting time (time to reach the most probable configuration) yields a bound on the mixing time, and if the minimum bias is a constant, then the hitting time is on the order of the area of the region.…”

Section: The Generalized Exclusion Markov Chain M Ementioning

confidence: 92%

“…Exchanging a 1 and a 0 corresponds to adding or removing a particular square beneath the staircase walk. Greenberg and others [9,20,10] considered sampling monotonic surfaces in Z 2 with bias. They studied walks that start at (0, h) and end at (w, 0) and only move to the right or down and analyze a "mountain / valley" chain that adds or removes a square along the boundary of the walk at each step (see Figure 1).…”

Section: The Generalized Exclusion Markov Chain M Ementioning

confidence: 99%

“…They studied walks that start at (0, h) and end at (w, 0) and only move to the right or down and analyze a "mountain / valley" chain that adds or removes a square along the boundary of the walk at each step (see Figure 1). In [10] and [20], they assumed the surface has "fluctuating bias," meaning that each square s (on the h × w lattice) is assigned a bias λ s which is essentially the ratio of the probabilities of adding or removing that particular square. They showed that as long as the minimum bias is a constant larger than 1 then the chain is rapidly mixing.…”

Section: The Generalized Exclusion Markov Chain M Ementioning

confidence: 99%

“…This continues to be an active area of interest and in recent work Labbé et al [14] determined the exact mixing rate and Levin et al [15] analyzed the case that p tends to 0 as n → ∞. Greenberg and others [20,10] considered a heterogeneous biased exclusion process, where the probability of swapping a 1 with a 0 at positions i and i + 1 depends on the number of 1's and the number of 0's to the left of position i.…”

Section: Introductionmentioning

confidence: 99%

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Rapid Mixing of k-Class Biased Permutations

Miracle

Streib

2018

LATIN 2018: Theoretical Informatics

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In this paper, we study a biased version of the nearest-neighbor transposition Markov chain on the set of permutations where neighboring elements i and j are placed in order (i, j) with probability p i,j . Our goal is to identify the class of parameter sets P = {p i,j } for which this Markov chain is rapidly mixing. Specifically, we consider the open conjecture of Jim Fill that all monotone, positively biased distributions are rapidly mixing.We resolve Fill's conjecture in the affirmative for distributions arising from k-class particle processes, where the elements are divided into k classes and the probability of exchanging neighboring elements depends on the particular classes the elements are in. We further require that k is a constant, and all probabilities between elements in different classes are bounded away from 1/2. These particle processes arise in the context of self-organizing lists and our result also applies beyond permutations to the setting where all particles in a class are indistinguishable. Additionally we show that a broader class of distributions based on trees is also rapidly mixing, which generalizes a class analyzed by Bhakta et. al. (SODA '13). Our work generalizes recent work by Haddadan and Winkler (STACS '17) studying 3-class particle processes.Our proof involves analyzing a generalized biased exclusion process, which is a nearestneighbor transposition chain applied to a 2-particle system. Biased exclusion processes are of independent interest, with applications in self-assembly. We generalize the results of Greenberg et al. (SODA '09) and Benjamini et. al (Trans. AMS '05) on biased exclusion processes to allow the probability of swapping neighboring elements to depend on the entire system, as long as the minimum bias is bounded away from 1.The fundamental problem of generating a random permutation has a long history in computer science, beginning as early as 1969 [13]. One way to generate a random permutation is to use the nearest-neighbor Markov chain M nn which repeatedly swaps the elements in a random pair of adjacent positions. The chain M nn was among the first considered in the study of the computational efficiency of Markov chains for sampling [1,6,4] and has subsequently been studied extensively. After a series of papers, the mixing time of M nn (Θ(n 3 log n) [25]) is now well-understood when sampling from the uniform distribution on the permutation group S n .The nearest-neighbor chain M nn can also be used to sample from more general probability distributions by allowing non-uniform swap probabilities. Suppose we have a set of parameters P = {p i,j } and that M nn puts neighboring elements i and j in order (i, j) with probability p i,j , where p j,i = 1 − p i,j . Despite the simplicity of this natural extension, much less is known about the mixing time of M nn in the non-uniform case. In this paper we look at the question for which parameter sets P is M nn rapidly (polynomially) mixing? We say P is positively

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Section: The Generalized Exclusion Markov Chain M Ementioning

confidence: 92%

Section: The Generalized Exclusion Markov Chain M Ementioning

confidence: 99%

Section: The Generalized Exclusion Markov Chain M Ementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Rapid Mixing of k-Class Biased Permutations

Miracle

Streib

2018

LATIN 2018: Theoretical Informatics

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“…In [10,6,8,27,28,47], the case of interfaces interacting with a substrate has been considered. The references [11,8,16,9] investigate the mixing time of higher dimensional interfaces. In [5], interfaces with real valued height functions are considered beyond the case of the random walk on the simplex.…”

Section: More General Interfacesmentioning

confidence: 99%

Mixing time and cutoff for one dimensional particle systems

Lacoin¹

2021

Preprint

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We survey recent results concerning the total-variation mixing time of the simple exclusion process on the segment (symmetric and asymmetric) and a continuum analog, the simple random walk on the simplex with an emphasis on cutoff results. A Markov chain is said to exhibit cutoff if on a certain time scale, the distance to equilibrium drops abruptly from 1 to 0. We also review a couple of techniques used to obtain these results by exposing and commenting some elements of proof.

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Mixing times of Markov chains for self‐organizing lists and biased permutations

Bhakta

Miracle

Randall

et al. 2022

Random Struct Algorithms

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We study the mixing time of a Markov chain on biased permutations, a problem related to self‐organizing lists. We are given probabilities false{pi,jfalse},$$ \left\{{p}_{i,j}\right\}, $$ for all i≠j,$$ i\ne j, $$ such that pi,j=1prefix−pj,i$$ {p}_{i,j}=1-{p}_{j,i} $$. The chain ℳnn$$ {\mathcal{M}}_{nn} $$ iteratively chooses two adjacent elements i$$ i $$ and j$$ j $$, and swaps them with probability pi,j$$ {p}_{i,j} $$. It has been conjectured that ℳnn$$ {\mathcal{M}}_{nn} $$ is rapidly mixing whenever the set of probabilities are “positively biased,” that is, false{pi,j≥1false/2false},$$ \left\{{p}_{i,j}\ge 1/2\right\}, $$ for all i

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Sampling biased monotonic surfaces using exponential metrics

Cited by 6 publications

References 22 publications

Rapid Mixing of k-Class Biased Permutations

Rapid Mixing of k-Class Biased Permutations

Mixing time and cutoff for one dimensional particle systems

Mixing times of Markov chains for self‐organizing lists and biased permutations

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