Statistical mechanics of combinatorial partitions, and their limit shapes

Vershik, A. M.

doi:10.1007/bf02509449

Cited by 223 publications

(309 citation statements)

References 5 publications

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“…Partitions into perfect squares turned out to be of interest in statistical mechanics as they have been used to model ideal gas. We refer to Vershik [39,Sec. 3,item 5.]…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

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Random partitions with restricted part sizes

Goh¹,

Hitczenko²

2007

Random Struct Algorithms

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ABSTRACT:For a subset S of positive integers let (n, S) be the set of partitions of n into summands that are elements of S. For every λ ∈ (n, S), let M n (λ) be the number of parts, with multiplicity, that λ has. Put a uniform probability distribution on (n, S), and regard M n as a random variable. In this paper the limiting density of the (suitably normalized) random variable M n is determined for sets that are sufficiently regular. In particular, our results cover the case S = {Q(k) : k ≥ 1}, where Q(x) is a fixed polynomial of degree d ≥ 2. For specific choices of Q, the limiting density has appeared before in rather different contexts such as Kingman's coalescent, and processes associated with the maxima of Brownian bridge and Brownian meander processes.

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“…Partitions into perfect squares turned out to be of interest in statistical mechanics as they have been used to model ideal gas. We refer to Vershik [39,Sec. 3,item 5.]…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…Partitions whose parts are restricted to a subset S ⊂ N correspond to b k := I k∈S , k ≥ 1, but other sequences (for example, b k = k α , α ≥ 0) have been considered, particularly in statistical physics (see [39] and also [16,Sec. 6] for further discussion).…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Random partitions with restricted part sizes

Goh¹,

Hitczenko²

2007

Random Struct Algorithms

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“…Consider the uniform measure on the set of all partitions of N. Then, as N → ∞, the diagrams of typical partitions are concentrated, see [32] for more precise statements, in the neighborhood of the shape bounded by the following curve exp − ζ(2) N x + exp − ζ(2) N y = 1 .…”

Section: The Case Of Simple Branched Coveringsmentioning

confidence: 99%

Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials

Eskin

Okounkov²

2001

Inventiones Mathematicae

154

216

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“…Remark 1.2. It should be stressed that our universality result is in sharp contrast with the onedimensional case, where the limit shape of random Young diagrams heavily depends on the distributional type [4,10,20,22]. Thus, the limit shape of (strict) vector partitions is a relatively "soft" property; such a distinction is essentially due to the different ways of geometrization used in the two models (i.e., convex polygonal lines vs. Young diagrams), resulting in similar but not identical functionals responsible for the limit shape (cf.…”

Section: Introductionmentioning

confidence: 57%

Limit shape of random convex polygonal lines: Even more universality

Bogachev

2014

Journal of Combinatorial Theory, Series A

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The paper concerns the limit shape (under some probability measure) of convex polygonal lines with vertices on Z 2 + , starting at the origin and with the right endpoint n = (n 1 , n 2 ) → ∞. In the case of the uniform measure, an explicit limit shape γ * := {(x 1 , x 2 ) ∈ R 2 + : √ 1 − x 1 + √ x 2 = 1} was found independently by Vershik (1994), Bárány (1995), andSinai (1994). Recently, Bogachev and Zarbaliev (2011) proved that the limit shape γ * is universal for a certain parametric family of multiplicative probability measures generalizing the uniform distribution. In the present work, the universality result is extended to a much wider class of multiplicative measures, including (but not limited to) analogs of the three meta-types of decomposable combinatorial structures -multisets, selections, and assemblies. This result is in sharp contrast with the one-dimensional case where the limit shape of Young diagrams associated with integer partitions heavily depends on the distributional type.

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Statistical mechanics of combinatorial partitions, and their limit shapes

Cited by 223 publications

References 5 publications

Random partitions with restricted part sizes

Random partitions with restricted part sizes

Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials

Limit shape of random convex polygonal lines: Even more universality

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