TY - JOUR

T1 - Gap probability for products of random matrices in the critical regime

AU - Berezin, Sergey

AU - Strahov, Eugene

N1 - Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2022/2

Y1 - 2022/2

N2 - The singular values of a product of M independent Ginibre matrices of size N×N form a determinantal point process. Near the soft edge, 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,+∞). We derive a Tracy–Widom-like formula in terms of the unique solution of a certain matrix Riemann–Hilbert problem of size 2 × 2. The right-tail asymptotics for this solution is obtained by the Deift–Zhou non-linear steepest descent analysis.

AB - The singular values of a product of M independent Ginibre matrices of size N×N form a determinantal point process. Near the soft edge, 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,+∞). We derive a Tracy–Widom-like formula in terms of the unique solution of a certain matrix Riemann–Hilbert problem of size 2 × 2. The right-tail asymptotics for this solution is obtained by the Deift–Zhou non-linear steepest descent analysis.

KW - Determinantal point processes

KW - Gap probabilities

KW - Products of random matrices

KW - Riemann–Hilbert problems

KW - Singular value statistics

UR - http://www.scopus.com/inward/record.url?scp=85121378433&partnerID=8YFLogxK

U2 - 10.1016/j.jat.2021.105687

DO - 10.1016/j.jat.2021.105687

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AN - SCOPUS:85121378433

SN - 0021-9045

VL - 274

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

M1 - 105687

ER -