Gap probability for products of random matrices in the critical regime

Sergey Berezin, Eugene Strahov

Research output: Contribution to journalArticlepeer-review


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.

Original languageAmerican English
Article number105687
JournalJournal of Approximation Theory
StatePublished - Feb 2022


  • Determinantal point processes
  • Gap probabilities
  • Products of random matrices
  • Riemann–Hilbert problems
  • Singular value statistics

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics
  • Numerical Analysis
  • General Mathematics


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