Product Matrix Processes as Limits of Random Plane Partitions

Borodin, Alexei; Gorin, Vadim; Strahov, Eugene

doi:10.1093/imrn/rny297

Cited by 16 publications

(16 citation statements)

References 43 publications

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“…The latter has been shown by now to be universal for a wide range of potentials, see [13] and references therein. One of our main tool, principally specialized Schur processes [16], has been used on other occasions [6,4] to bridge algebraic combinatorics and random matrix theory. Finally, let us note that Thm.…”

Section: Main Contributionmentioning

confidence: 99%

Muttalib--Borodin plane partitions and the hard edge of random matrix ensembles

Betea,

Occelli

2020

Preprint

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We study probabilistic and combinatorial aspects of natural volume-andtrace weighted plane partitions and their continuous analogues. We prove asymptotic limit laws for the largest parts of these ensembles in terms of new and known hardand soft-edge distributions of random matrix theory. As a corollary we obtain an asymptotic transition between Gumbel and Tracy-Widom GUE fluctuations for the largest part of such plane partitions, with the continuous Bessel kernel providing the interpolation. We interpret our results in terms of two natural models of directed last passage percolation (LPP): a discrete (max, +) infinite-geometry model with rapidly decaying geometric weights, and a continuous (min, •) model with power weights.

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Section: Main Contributionmentioning

confidence: 99%

Muttalib--Borodin plane partitions and the hard edge of random matrix ensembles

Betea,

Occelli

2020

Preprint

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“…A manifestation of the universality and fundamental importance of these limiting determinantal processes lies in the fact that they are also scaling limits of combinatorial models without a priori relation to random matrices, much like it is the case for the classical Airy, Bessel, and sine processes. For further information see Ahn [1], and Borodin, Gorin and Strahov [12].…”

Section: Introductionmentioning

confidence: 99%

Gap probability for products of random matrices in the critical regime

Berezin,

Strahov

2021

Preprint

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The singular values of a product of M independent Ginibre matrices of size N ×N form a determinantal point process. As both M and N go to infinity in such a way that M/N → α, α > 0, a scaling limit emerges. We consider a gap probability for the corresponding limiting determinantal process, namely, the probability that there are no particles in the interval (a, +∞), a > 0. This probability is evaluated explicitly in terms of the unique solution of a certain matrix Riemann-Hilbert problem of size 2 × 2. The right-tail asymptotics of this solution is obtained by the Deift-Zhou non-linear steepest descent analysis.

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“…These densities have explicit determinantal forms in terms of Meijer G-functions, which leads to determinantal point processes formed by the squared singular values. Through the remarkable fact that the product matrix process with truncated unitary matrices can be understood as a scaling limit of the Schur process, see Borodin, Gorin, and Strahov [6], one can obtain determinantal formulas for (dynamical) correlation functions.…”

Section: Introductionmentioning

confidence: 99%

“…More concretely, the squared singular values of these products can be realized as a degeneration of certain Macdonald processes. In [6], this degeneration was applied in the unitary case, and [4] considered this degeneration for the other symmetry classes. Through this connection, we produce an explicit formula for the Markov transition kernel from squared singular value of a deterministic matrix X to the squared singular values of a product T X where T is a truncated orthogonal or symplectic matrix, analogous to the result of [14,15].…”

Section: Introductionmentioning

confidence: 99%

Product Matrix Processes with Symplectic and Orthogonal Invariance via Symmetric Functions

Ahn,

Strahov

2020

Preprint

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We apply the theory of symmetric functions to product matrix processes formed by singular values of products of truncations of Haar distributed symplectic and orthogonal matrices. These product matrix processes can be understood as scaling limits of Macdonald processes introduced by Borodin and Corwin. The relation with Macdonald processes enables us to derive the distribution of the singular values of a fixed matrix multiplied by a truncation of a Haar distributed orthogonal or symplectic matrix. As a consequence we obtain explicit formulae for the distributions of singular values of the product matrix processes under considerations, and, in particular, generalize the recent Kieburg-Kuijlaars-Stivigny formula for the joint singular value density of a product of truncated unitary matrices to symplectic and orthogonal symmetry classes. We apply these results to products of two symplectic matrices, and prove that the squared singular values of such products form a Pfaffian process. We derive explicit formulae for the correlation kernel of this process.

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Product Matrix Processes as Limits of Random Plane Partitions

Cited by 16 publications

References 43 publications

Muttalib--Borodin plane partitions and the hard edge of random matrix ensembles

Muttalib--Borodin plane partitions and the hard edge of random matrix ensembles

Gap probability for products of random matrices in the critical regime

Product Matrix Processes with Symplectic and Orthogonal Invariance via Symmetric Functions

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