Two faces of Douglas-Kazakov transition: from Yang-Mills theory to random walks and beyond

TL;DR

This paper explores the Douglas-Kazakov phase transition across gauge theories and stochastic processes, revealing new connections and physical interpretations, including implications for black hole physics and the influence of initial conditions and parameters.

Contribution

It generalizes the YM-VW connection, studies the impact of initial distributions and the $ heta$-term, and links the DK transition to black hole physics through $q$-YM.

Findings

Influence of initial and final distributions on DK transition

Effect of the $ heta$-term in stochastic processes

Dependence of DK transition on coupling constants

Abstract

Being inspired by the connection between 2D Yang-Mills (YM) theory and (1+1)D "vicious walks" (VW), we consider different incarnations of large-$N$ Douglas-Kazakov (DK) phase transition in gauge field theories and stochastic processes focusing at possible physical interpretations. We generalize the connection between YM and VW, study the influence of initial and final distributions of walkers on the DK phase transition, and describe the effect of the $\theta$-term in corresponding stochastic processes. We consider the Jack stochastic process involving Calogero-type interaction between walkers and investigate the dependence of DK transition point on a coupling constant. Relying on the relation between large-$N$ 2D $q$-YM and extremal black hole (BH) with large-$N$ magnetic charge, we speculate about a physical interpretation of a DK phase transitions in a 4D extremal charged BH.