Double-scaling limits of Toeplitz determinants with merging Fisher-Hartwig singularity on the unit circle

Release time:2025-12-13

Journal:Submitted

Abstract:We study the transition asymptotics of Toeplitz determinants with symbols \(f_{t}\) depending on a parameter $t$ where \(f_{t}\) has three Fisher-Hartwig singularities when \(t>0\) and two Fisher-Hartwig singularities when $t=0$. Using the Riemann-Hilbert approach to orthogonal polynomials, we establish double scaling limits where \(n \to \infty\) and simultaneously \(t \to 0\), and express the transition in terms of Painlevé transcendents. The results extend previous analysis of merging singularities and highlight a new type of Fisher–Hartwig transition on the unit circle. As an application, we study gap probability for thinned Circle Unitary Ensemble with an external potential.

Co-author:Pan Ma, Xuanzhuo Zhou

Indexed by:Journal paper

Discipline:Natural Science

First-Level Discipline:Mathematics

Document Type:J

Translation or Not:no