Fluctuations of particle systems determined by Schur generating functions

Bufetov, Alexander I.; Gorin, Vadim

doi:10.1016/j.aim.2018.07.009

Cited by 71 publications

(110 citation statements)

References 45 publications

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“…These fluctuations have been studied in detail in the free fermion case in Refs. [34][35][36][37][38], with the conclusion that the fluctuations are always given by a Gaussian Free Field (GFF) [39] in a certain metric (we will discuss the metric shortly). This problem was revisited recently by some of us and our collaborators [40], also for free fermions, motivated by connections with out-of-equilibrium problems in one-dimensional quantum systems [41,42].…”

Section: Figurementioning

confidence: 99%

Inhomogeneous Gaussian free field inside the interacting arctic curve

Granet¹,

Budzynski²,

Dubail³

et al. 2019

J. Stat. Mech.

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The six-vertex model with domain-wall boundary conditions is one representative of a class of two-dimensional lattice statistical mechanics models that exhibit a phase separation known as the arctic curve phenomenon. In the thermodynamic limit, the degrees of freedom are completely frozen in a region near the boundary, while they are critically fluctuating in a central region. The arctic curve is the phase boundary that separates those two regions. Critical fluctuations inside the arctic curve have been studied extensively, both in physics and in mathematics, in free models (i.e., models that map to free fermions, or equivalently to determinantal point processes). Here we study those critical fluctuations in the interacting (i.e., not free, not determinantal) six-vertex model, and provide evidence for the following two claims:(i) the critical fluctuations are given by a Gaussian Free Field (GFF), as in the free case, but (ii) contrarily to the free case, the GFF is inhomogeneous, meaning that its coupling constant K becomes position-dependent, K → K(x).The evidence is mainly based on the numerical solution of appropriate Bethe ansatz equations with an imaginary extensive twist, and on transfer matrix computations, but the second claim is also supported by the analytic calculation of K and its first two derivatives in selected points. Contrarily to the usual GFF, this inhomogeneous GFF is not defined in terms of the Green's function of the Laplacian ∆ = ∇ · ∇ inside the critical domain, but instead, of the Green's function of a generalized Laplacian ∆ = ∇ · 1 K ∇ parametrized by the function K. Surprisingly, we also find that there is a change of regime when ∆ ≤ −1/2, with K becoming singular at one point.

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Section: Figurementioning

confidence: 99%

Inhomogeneous Gaussian free field inside the interacting arctic curve

Granet¹,

Budzynski²,

Dubail³

et al. 2019

J. Stat. Mech.

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show abstract

“…(49)). The relation (22) was proved by Erdős and Schröder, in greater generality (Wigner matrices) and stronger topology (corresponding to test functions in the Sobolev space H 2 [−10, 10]); they used it to prove (15). The relation (21) seems not to have been observed before.…”

Section: Fluctuations About the Limiting Shapementioning

confidence: 90%

“…Another proof, based on Kerov's unpublished notes, was given by Ivanov and Olshanski in [25]. We refer to the works [15,44,45] for various generalisations, not discussed here. Ivanov and Olshanski also described the fluctuations of the transition measurẽ µ n associated with ω n (see 4.1.3):…”

Section: Fluctuations About the Limiting Shapementioning

confidence: 99%

“…Differentiating the relations (19)- (20) and comparing to (15)- (16), we obtain an answer in the following form Figure 2: The random continual diagrams (19) in blue and (15) in green, for GUE n=50 . The former fluctuates on scale ∼ 1/n, while the latter -on scale 1/ √ n.…”

Section: Fluctuations About the Limiting Shapementioning

confidence: 99%

“…As mentioned in Section 1.1 above, the Markov transform takes the left-hand side of (15) to the spectral measureμ n (see 2.3.4). The fluctuations of the latter were studied, in the context of Gaussian ensembles, by Lytova and Pastur [41]; for the GUE, their result (see 3.2.3) asserts that…”

Section: Fluctuations About the Limiting Shapementioning

confidence: 99%

See 2 more Smart Citations

Fluctuations of Interlacing Sequences

Sodin¹

2017

Z. mat. fiz. anal. geom.

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In a series of works published in the 1990-s, Kerov put forth various applications of the circle of ideas centred at the Markov moment problem to the limiting shape of random continual diagrams arising in representation theory and spectral theory. We demonstrate on several examples that his approach is also adequate to study the fluctuations about the limiting shape.In the random matrix setting, we compare two continual diagrams: one is constructed from the eigenvalues of the matrix and the critical points of its characteristic polynomial, whereas the second one is constructed from the eigenvalues of the matrix and those of its principal submatrix. The fluctuations of the latter diagram were recently studied by Erdős and Schröder; we discuss the fluctuations of the former, and compare the two limiting processes.For Plancherel random partitions, the Markov correspondence establishes the equivalence between Kerov's central limit theorem for the Young diagram and the Ivanov-Olshanski central limit theorem for the transition measure. We outline a combinatorial proof of the latter, and compare the limiting process with the ones of random matrices.

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Rigidity and Edge Universality of Discrete β‐Ensembles

Guionnet

Huang

2019

Comm Pure Appl Math

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We study discrete β‐ensembles as introduced in [17]. We obtain rigidity estimates on the particle locations; i.e., with high probability, the particles are close to their classical locations with an optimal error estimate. We prove the edge universality of the discrete β‐ensembles; i.e., for β ≥ 1, the distribution of extreme particles converges to the Tracy‐Widom β‐distribution. As far as we know, this is the first proof of general Tracy‐Widom β‐distributions in the discrete setting. A special case of our main results implies that under the Jack deformation of the Plancherel measure, the distribution of the lengths of the first few rows in Young diagrams converges to the Tracy‐Widom β‐distribution, which answers an open problem in [38]. Our proof relies on Nekrasov's (or loop) equations, a multiscale analysis, and a comparison argument with continuous β‐ensembles. © 2019 Wiley Periodicals, Inc.

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Fluctuations of particle systems determined by Schur generating functions

Cited by 71 publications

References 45 publications

Inhomogeneous Gaussian free field inside the interacting arctic curve

Inhomogeneous Gaussian free field inside the interacting arctic curve

Fluctuations of Interlacing Sequences

Rigidity and Edge Universality of Discrete β‐Ensembles

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