Universality properties of Gelfand–Tsetlin patterns

Metcalfe, Anthony

doi:10.1007/s00440-011-0399-7

Cited by 25 publications

(32 citation statements)

References 27 publications

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“…Perhaps the most similar minor process to ours is that studied by the author Metcalfe in [22]. There, the eigenvalues of A n are deterministic: λ (n) = x (n) for some fixed x (n) ∈ R n with x (n) 1 > x (n) 2 > · · · > x (n) n , and the eigenvalue minor process induces the uniform probability distribution on the set of Gelfand-Tsetlin patterns of depth n with the particles on the top row in the deterministic positions defined by x (n) ∈ R n .…”

supporting

confidence: 75%

“…for all n sufficiently large. Indeed, the second part follows since H n = Z n \ P n (see equation (22)), and it implies that particles are eventually densely packed in R λ−µ ( ).…”

mentioning

confidence: 99%

“…for all n ≥ 1. Assumption 1.1 and equation (22) then imply that µ n → µ weakly as n → ∞. Next, let C n : C \ Supp(µ n ) → C denote the Cauchy Transform of µ n : (24)), and since R µ ( ) + ∪ R µ ( ) − and R λ−µ ( ) are disjoint (see equation (23)).…”

mentioning

confidence: 99%

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Universal edge fluctuations of discrete interlaced particle systems

Duse¹,

Metcalfe²

2018

Annales Mathématiques Blaise Pascal

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We impose the uniform probability measure on the set of all discrete Gelfand-Tsetlin patterns of depth n with the particles on row n in deterministic positions. These systems equivalently describe a broad class of random tilings models, and are closely related to the eigenvalue minor processes of a broad class of random Hermitian matrices. They have a determinantal structure, with a known correlation kernel. We rescale the systems by 1 n , and examine the asymptotic behaviour, as n → ∞, under weak asymptotic assumptions for the (rescaled) particles on row n: The empirical distribution of these converges weakly to a probability measure with compact support, and they otherwise satisfy mild regulatory restrictions.We prove that the correlation kernel of particles in the neighbourhood of 'typical edge points' convergences to the extended Airy kernel. To do this, we first find an appropriate scaling for the fluctuations of the particles. We give an explicit parameterisation of the asymptotic edge, define an analogous non-asymptotic edge curve (or finite n-deterministic equivalent), and choose our scaling such that that the particles fluctuate around this with fluctuations of order O(n − 1 3 ) and O(n − 2 3 ) in the tangent and normal directions respectively. While the final results are quite natural, the technicalities involved in studying such a broad class of models under such weak asymptotic assumptions are unavoidable and extensive.

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supporting

confidence: 75%

“…for all n sufficiently large. Indeed, the second part follows since H n = Z n \ P n (see equation (22)), and it implies that particles are eventually densely packed in R λ−µ ( ).…”

mentioning

confidence: 99%

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Universal edge fluctuations of discrete interlaced particle systems

Duse¹,

Metcalfe²

2018

Annales Mathématiques Blaise Pascal

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“…Furthermore, it is known that the asymptotic shapes of interfaces with the bead interaction potential have intimate relationships with models in free probability (see e.g. Metcalfe [26]). With these observations in mind, and in the event that Conjecture 4.2 holds, we are lead to further predict the following result about the asymptotic shape of a certain class of interfaces.…”

Section: Scaling Limits Of Stochastic Interfaces 41 Surface Tension mentioning

confidence: 99%

Scaling Limits for Non-intersecting Polymers and Whittaker Measures

Johnston

O’Connell

2020

J Stat Phys

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We study the partition functions associated with non-intersecting polymers in a random environment. By considering paths in series and in parallel, the partition functions carry natural notions of subadditivity, allowing the effective study of their asymptotics. For a certain choice of random environment, the geometric RSK correspondence provides an explicit representation of the partition functions in terms of a stochastic interface. Formally this leads to a variational description of the macroscopic behaviour of the interface and hence the free energy of the associated non-intersecting polymer model. At zero temperature we relate this variational description to the Marčenko-Pastur distribution, and give a new derivation of the surface tension of the bead model.

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“…The proof given here is that in [35]. A corresponding result for continuous interlacing particle systems was studied in [60], see also [30]. Remark 5.3.…”

Section: It Follows That Detmentioning

confidence: 82%

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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Universality properties of Gelfand–Tsetlin patterns

Cited by 25 publications

References 27 publications

Universal edge fluctuations of discrete interlaced particle systems

Universal edge fluctuations of discrete interlaced particle systems

Scaling Limits for Non-intersecting Polymers and Whittaker Measures

Edge fluctuations of limit shapes

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