Limit shapes, real and imagined

Okounkov, Andreĭ

doi:10.1090/bull/1512

Cited by 30 publications

(43 citation statements)

References 40 publications

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“…In a sense, this issue is complementary to a large body of works, see for instance [16,14,22,30] and references to later papers, and also a recent review [32], which focus on study of the corners of zero or low temperature microscopic crystals.…”

Section: Introductionmentioning

confidence: 89%

Formation of Facets for an Effective Model of Crystal Growth

Ioffe¹,

Shlosman²

2019

Springer Proceedings in Mathematics &Amp; Statistics

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We study an effective model of microscopic facet formation for low temperature three dimensional microscopic Wulff crystals above the droplet condensation threshold. The model we consider is a 2+1 solid on solid surface coupled with high and low density bulk Bernoulli fields. At equilibrium the surface stays flat. Imposing a canonical constraint on excess number of particles forces the surface to "grow" through the sequence of spontaneous creations of macroscopic size monolayers. We prove that at all sufficiently low temperatures, as the excess particle constraint is tuned, the model undergoes an infinite sequence of first order transitions, which traces an infinite sequence of first order transitions in the underlying variational problem. Away from transition values of canonical constraint we prove sharp concentration results for the rescaled level lines around solutions of the limiting variational problem.

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Section: Introductionmentioning

confidence: 89%

Formation of Facets for an Effective Model of Crystal Growth

Ioffe¹,

Shlosman²

2019

Springer Proceedings in Mathematics &Amp; Statistics

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“…A perfect matching is equivalent to a tiling of R 2 by three types of lozenges, one type for each of the three edges incident to each vertex. Using a natural height function, such a lozenge tiling gives a surface, and the study of the statistical properties of such random surfaces has resulted in many remarkable discoveries (see Okounkov's survey [29] or Gorin's detailed account of lozenge tilings [16]). Kasteleyn discovered that by cleverly assigning signs to the edges of H, he could compute the number of perfect matchings on a finite approximation using periodic boundary conditions by a determinant formula.…”

Section: Examplesmentioning

confidence: 99%

Decimation limits of principal algebraic $\mathbb{Z}^d$-actions

Arzhakova¹,

Lind²,

Schmidt³

et al. 2021

Preprint

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Let f be a Laurent polynomial in d commuting variables with integer coefficients. Associated to f is the principal algebraic Z d -action α f on a compact subgroup X f of T Z d determined by f . Let N 1 and restrict points in X f to coordinates in N Z d . The resulting algebraic N Z d -action is again principal, and is associated to a polynomial gN whose support grows with N and whose coefficients grow exponentially with N . We prove that by suitably renormalizing these decimations we can identify a limiting behavior given by a continuous concave function on the Newton polytope of f , and show that this decimation limit is the negative of the Legendre dual of the Ronkin function of f . In certain cases with two variables, the decimation limit coincides with the surface tension of random surfaces related to dimer models, but the statistical physics methods used to prove this are quite different and depend on special properties of the polynomial.

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“…with equality if and only if in the expansion (18), d j = 0, for all j : q |j, for some integer q ≥ 2. This says that in the case of equality, the condition (33) of Theorem 2 does not hold, which leads to |φ n ( p q )| = 1, in contradiction to the condition (68).…”

Section: • Theoremmentioning

confidence: 99%

Asymptotic enumeration by Khintchine–Meinardus probabilistic method: necessary and sufficient conditions for sub-exponential growth

Granovsky

2017

Ramanujan J

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In this paper we prove the necessity of the main sufficient condition of Meinardus for sub exponential rate of growth of the number of structures, having multiplicative generating functions of a general form and establish a new necessary and sufficient condition for normal local limit theorem for aforementioned structures. The latter result allows to encompass in our study structures with sequences of weights having gaps in their support.

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Limit shapes, real and imagined

Abstract: Abstract. This is an introductory discussion of limit shapes, in particular for random partitions and stepped surfaces, and of their applications to supersymmetric gauge theories.

Cited by 30 publications

References 40 publications

Formation of Facets for an Effective Model of Crystal Growth

Formation of Facets for an Effective Model of Crystal Growth

Decimation limits of principal algebraic $\mathbb{Z}^d$-actions

Asymptotic enumeration by Khintchine–Meinardus probabilistic method: necessary and sufficient conditions for sub-exponential growth

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