The Cusp-Airy process

Duse, Erik; Johansson, Kurt; Metcalfe, Anthony

doi:10.1214/16-ejp2

Cited by 21 publications

(50 citation statements)

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“…in [19,20,2,1,5]. This paper is based and is a follow-up of papers [3,4]; see also [6,9,16,22,23] To find out such a statistics for the lozenge tilings, we first need to determine the correlation kernel for the finite problem, in particular showing that the point process of blue tiles (replaced by blue dots in the middle of the tiles) along the parallel lines, mentioned above, is determinantal, with an explicitly given correlation kernel K blue (ξ 1 , η 1 ; ξ 2 , η 2 ), in the variables η, ξ (to be explained later). For convenience of coordinates, we perform an affine transformation of Fig.…”

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confidence: 91%

Probability distributions related to tilings of non-convex polygons

Adler

Moerbeke

2018

Journal of Mathematical Physics

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This paper is based on the study of random lozenge tilings of non-convex polygonal regions with interacting non-convexities (cuts) and the corresponding asymptotic kernel as in [3] and [4] (discrete tacnode kernel). Here this kernel is used to the find the probability distributions and joint probability distributions for the fluctuation of tiles along lines in between the cuts. These distributions are new. Dedicated to the memory of Ludvig Faddeev1 The discrete tacnode kernel and main result Domino or lozenge tilings of large geometric shapes constitute a rich source of new statistical phenomena: they have sufficient complexity to have interesting features and yet are simple enough to be tractable! Most models studied sofar display two phases, a solid phase with a bricklike pattern, and a liquid phase, for which the correlations decay polynomially with distance. More recently, new models were considered having an additional phase, a gas phase, for which the correlations decay exponentially with distance. This paper will deal with a model having the two phases, solid and liquid.A celebrated example goes back to MacMahon [21] in 1911, who found a simple combinatorial formula for the number of lozenge tilings of a hexagon of sides a, b, c, a, b, c. This model has been widely studied and extended from the macroscopic point of view, but also from the microscopic point of view [8,12,13,7]. For tilings of hexagons, everything is known: when the size gets large, an arctic ellipse, inscribed in the hexagon, separates the liquid phase and the solid phases appearing in the six corners of the hexagon. The liquid phase behaves like a Gaussian Free Field, the statistical fluctuations of the tiles along the ellipse fluctuate according to the Airy process [14]. The tiles in the neighborhood of the tangency points of the arctic ellipse with the hexagon behave as the eigenvalues of the consecutive principal minors of a GUE-matrix (GUE-minor process) [15]. They are all universal distributions, in the sense that they have been found in entirely different circumstances as well. They are also known to occur at critical points along the boundary between phases. The universal distributions are "integrable": many of them relate to known integrable systems, like KdV equation, the Boussinesq equation, Toda lattices, etc... or they can be treated by means of Riemann-Hilbert methods.This paper written in memory of Ludvig Faddeev is a tribute to his pioneering contributions to the integrable field. In 1970-71, he gave a lecture at Rockefeller University in NY on the KdV equation, showing that KdV is a completely integrable Hamiltonian system, and that the map to the spectral data is symplectic. His brilliant lecture triggered the interest and inspiration of one of the authors of this paper (PvM): thank you, Ludvig! Domino tilings of Aztec diamonds have also been extensively studied from the combinatorial point of view and from the microscopic point of view [10,11,18,24,12,14].

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confidence: 91%

Probability distributions related to tilings of non-convex polygons

Adler

Moerbeke

2018

Journal of Mathematical Physics

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“…This kernel can be established as a limit of a model of interlacing particles, or rhombus tilings, as discussed in section 5 as was done in [34]. This type of situation was first studied in [66] under the name cuspidal turning point in skew plane partitions but the kernel given there has the wrong contours and no proof was given.…”

Section: Scaling Limitsmentioning

confidence: 99%

Edge fluctuations of limit shapes

Johansson

2016

Current Developments in Mathematics

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In random tiling and dimer models we can get various limit shapes which gives the boundaries between different types of phases. The shape fluctuations at these boundaries give rise to universal limit laws, in particular the Airy process. We survey some models which can be analyzed in detail based on the fact that they are determinantal point processes with correlation kernels that can be computed. We also discuss which type of limit laws that can be obtained. 16 3.

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“…and the tacnode-GUE statistics [1,4]; it should undoubtedly also lead to Pearcey statistics. Random lozenge tilings of polygons with cuts have led to the Cusp-Airy statistics [29,13]. All of these phenomena appear as result of non-convexities or cuts in the regions.…”

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confidence: 99%

“…4, satisfying m 1 + m 2 = n 1 + n 2 and N = b L + c L = b R + c R and c L + d = c R + d , the same limit (14) holds. Here also the limiting kernel L dTac , as in (9), only depends on the width ρ of the oblique strip {ρ } formed by the two cuts, the number r of dots on each oblique line in the strip {ρ }, and β as in (13),where…”

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confidence: 99%