Problem Solving is Often a Matter of Cooking Up an Appropriate Markov Chain*

Häggström, Olle

doi:10.1111/j.1467-9469.2007.00561.x

Cited by 6 publications

(7 citation statements)

References 30 publications

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“…The technique of proof has interest from a probabilistic point of view, because it analyses a measure by regarding it as the equilibrium of a random process that is introduced for this purpose. Arguments in which a convenient Markov chain is defined and analysed have previously been used to prove such results as correlation inequalities in percolation [23]: see [13] for a review. Of course, the Metropolis algorithm is very commonly used to sample approximately a measure (as reviewed in Chapter 3 of [19]).…”

Section: Techniques and Relationsmentioning

confidence: 99%

Phase separation in random cluster models II: The droplet at equilibrium, and local deviation lower bounds

Hammond¹

2012

Ann. Probab.

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We study the droplet that results from conditioning the planar subcritical Fortuin-Kasteleyn random cluster model on the presence of an open circuit Γ 0 encircling the origin and enclosing an area of at least (or exactly) n 2 . We consider local deviation of the droplet boundary, measured in a radial sense by the maximum local roughness, MLR Γ 0 , this being the maximum distance from a point in the circuit Γ 0 to the boundary ∂conv Γ 0 of the circuit's convex hull; and in a longitudinal sense by what we term maximum facet length, namely, the length of the longest line segment of which the polygon ∂conv Γ 0 is formed. We prove that there exists a constant c > 0 such that the conditional probability that the normalised quantity n −1/3 log n −2/3 MLR Γ 0 exceeds c tends to 1 in the high n-limit; and that the same statement holds for n −2/3 log n −1/3 MFL Γ 0 . To obtain these bounds, we exhibit the random cluster measure conditional on the presence of an open circuit trapping high area as the invariant measure of a Markov chain that resamples sections of the circuit boundary. We analyse the chain at equilibrium to prove the local roughness lower bounds. Alongside complementary upper bounds provided in [14], the fluctuations MLR Γ 0 and MFL Γ 0 are determined up to a constant factor.

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Section: Techniques and Relationsmentioning

confidence: 99%

Phase separation in random cluster models II: The droplet at equilibrium, and local deviation lower bounds

Hammond¹

2012

Ann. Probab.

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“…By the main theorem in [5] or Theorem 6.B.9 in [4] whose alternate proof via coupling can be found in [6], it follows that…”

Section: )mentioning

confidence: 99%

Conditional ordering of order statistics

Zhuang

Yao

2010

Journal of Multivariate Analysis

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a b s t r a c tFor any positive integers m and n, let X 1 , X 2 , . . . , X m∨n be independent random variables with possibly nonidentical distributions. Let X 1:n ≤ X 2:n ≤ · · · ≤ X n:n be order statistics of random variables X 1 , X 2 , . . . , X n , and let X 1:m ≤ X 2:m ≤ · · · ≤ X m:m be order statistics of random variables X 1 , X 2 , . . . , X m . It is shown that (X j:n , X j+1:n , . . . , X n:n ) given X i:m > y for j − i ≥ max{n − m, 0}, and (X 1:n , X 2:n , . . . , X j:n ) given X i:m ≤ y for j − i ≤ min{n − m, 0} are all increasing in y with respect to the usual multivariate stochastic order. We thus extend the main results in Dubhashi and Häggström (2008) [1] and Hu and Chen (2008) [2].

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“…In this paper, we propose a modified FCC lattice to model a CNT fiber and employ the theory of finite Markov chains to direct at our purpose, so as to find out the percolation threshold of CNT fibers. Although various forms of Markov theory have been widely applied to percolation problems by setting up random fields, or characterizing percolation behavior, [42][43][44][45][46][47][48][49][50][51][52] it is barely used to check the occurrence of percolation. We 3 first describe our lattice model derived from FCC, which is known as the most closepacking structure in nature.…”

Section: Introductionmentioning

confidence: 99%

Site-percolation threshold of carbon nanotube fibers—Fast inspection of percolation with Markov stochastic theory

Yakobson

2014

Physica A: Statistical Mechanics and its Applications

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Abstract:We present a site-percolation model based on a modified FCC lattice, as well as an efficient algorithm of inspecting percolation which takes advantage of the Markov stochastic theory, in order to study the percolation threshold of carbon nanotube (CNT) fibers. Our Markov-chain based algorithm carries out the inspection of percolation by performing repeated sparse matrix-vector multiplications, which allows parallelized computation to accelerate the inspection for a given configuration. With this approach, we determine that the site-percolation transition of CNT fibers occurs at = 0.1533 ± 0.0013, and analyze the dependence of the effective percolation threshold (corresponding to 0.5 percolation probability) on the length and the aspect ratio of a CNT fiber on a finite-size-scaling basis. We also discuss the aspect ratio dependence of percolation probability with various values of p (not restricted to ).2

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Problem Solving is Often a Matter of Cooking Up an Appropriate Markov Chain*

Cited by 6 publications

References 30 publications

Phase separation in random cluster models II: The droplet at equilibrium, and local deviation lower bounds

Phase separation in random cluster models II: The droplet at equilibrium, and local deviation lower bounds

Conditional ordering of order statistics

Site-percolation threshold of carbon nanotube fibers—Fast inspection of percolation with Markov stochastic theory

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