Strong Stationary Times Via a New Form of Duality

Diaconis, Persi; Fill, James Allen

doi:10.1214/aop/1176990628

Cited by 226 publications

(427 citation statements)

References 28 publications

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“…[3], our Markov chain falls into the class of local growth models with relaxation and lateral growth, described by the Kardar-Parisi-Zhang (KPZ) equation 19) where Q is a quadratic form. Relations (1.10) and (1.11) imply that for our growth model the determinant of the Hessian of ∂ t h, viewed as a function of A configuration of the model analyzed with N = 100 particles at time t = 25, using the same representation as in Figure 1.2.…”

Section: Universality Classmentioning

confidence: 99%

“…Finally, let us mention that our proof of Theorem 1.1 is based on the argument of [18] and [50], the proof of Theorem 1.3 uses several ideas from [33], and the algebraic formalism for two-dimensional growth models employs a crucial idea of constructing bivariate Markov chains out of commuting univariate ones from [19].…”

Section: Other Connectionsmentioning

confidence: 99%

“…A characteristic feature of the construction is that x and y * are independent, given x * and y. This bivariate Markov chain is denoted P Λ ; its construction is borrowed from [19].…”

Section: Two Dimensional Dynamicsmentioning

confidence: 99%

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Anisotropic Growth of Random Surfaces in 2 + 1 Dimensions

Borodin

Ferrari

2013

Commun. Math. Phys.

221

548

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We construct a family of stochastic growth models in 2 + 1 dimensions, that belong to the anisotropic KPZ class. Appropriate projections of these models yield 1 + 1 dimensional growth models in the KPZ class and random tiling models. We show that correlation functions associated to our models have determinantal structure, and we study large time asymptotics for one of the models.The main asymptotic results are: (1) The growing surface has a limit shape that consists of facets interpolated by a curved piece.(2) The one-point fluctuations of the height function in the curved part are asymptotically normal with variance of order ln(t) for time t ≫ 1. (3) There is a map of the (2 + 1)-dimensional space-time to the upper half-plane H such that on space-like submanifolds the multipoint fluctuations of the height function are asymptotically equal to those of the pullback of the Gaussian free (massless) field on H.

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Section: Universality Classmentioning

confidence: 99%

Section: Other Connectionsmentioning

confidence: 99%

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Anisotropic Growth of Random Surfaces in 2 + 1 Dimensions

Borodin

Ferrari

2013

Commun. Math. Phys.

221

548

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“…Whenever we have an intertwining of (π 0 , P ) and (π 0 , P ), Section 2.4 of the strong stationary duality paper [9] by Diaconis and Fill gives a family of ways to create sample-path linking. Here is one [9, eq.…”

Section: 1mentioning

confidence: 99%

“…Then, for the bivariate process ( X, X), we see that absorption times agree: T 0 (X) = T0( X). For a parallel explanation of how sample-path linking can be used to connect the mixing time for an ergodic primary chain with a hitting time for a dual chain, consult [9]; very closely related is the FMMR perfect sampling algorithm [14,17].…”

Section: 1mentioning

confidence: 99%

Hitting Times and Interlacing Eigenvalues: A Stochastic Approach Using Intertwinings

Fill

Lyzinski

2012

J Theor Probab

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Abstract. We develop a systematic matrix-analytic approach, based on intertwinings of Markov semigroups, for proving theorems about hitting-time distributions for finite-state Markov chains-an approach that (sometimes) deepens understanding of the theorems by providing corresponding sample-path-by-sample-path stochastic constructions. We employ our approach to give new proofs and constructions for two theorems due to Mark Brown, theorems giving two quite different representations of hitting-time distributions for finite-state Markov chains started in stationarity. The proof, and corresponding construction, for one of the two theorems elucidates an intriguing connection between hitting-time distributions and the interlacing eigenvalues theorem for bordered symmetric matrices.

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Extension of Fill's perfect rejection sampling algorithm to general chains

Fill

Machida

Murdoch

et al. 2000

Random Struct. Alg.

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By developing and applying a broad framework for rejection sampling using auxiliary randomness, we provide an extension of the perfect sampling algorithm of Fill (1998) to general chains on quite general state spaces, and describe how use of bounding processes can ease computational burden. Along the way, we unearth a simple connection between the Coupling From The Past (CFTP) algorithm originated by Propp and Wilson (1996) and our extension of Fill's algorithm.

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Strong Stationary Times Via a New Form of Duality

Cited by 226 publications

References 28 publications

Anisotropic Growth of Random Surfaces in 2 + 1 Dimensions

Anisotropic Growth of Random Surfaces in 2 + 1 Dimensions

Hitting Times and Interlacing Eigenvalues: A Stochastic Approach Using Intertwinings

Extension of Fill's perfect rejection sampling algorithm to general chains

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