Critical behavior in topological ensembles

TL;DR

This paper explores the critical behavior across three physical systems—lattice paths, torus knots, and instanton ensembles—revealing new phase transition phenomena and explicit generating functions linked to $q$-Narayana numbers.

Contribution

It provides an explicit generating function for $q$-Narayana numbers and uncovers new critical behaviors and phase transitions in topological and gauge theory ensembles.

Findings

Critical behavior characterized by 'area+length+corners' statistics.

Explicit expression for the generating function of $q$-Narayana numbers.

Identification of different order phase transitions in various ensembles.

Abstract

We consider the relation between three physical problems: 2D directed lattice random walks, ensembles of $T_{n,n+1}$ torus knots, and instanton ensembles in 5D SQED with one compact dimension in $\Omega$ background and with 5D Chern-Simons term at the level one. All these ensembles exhibit the critical behavior typical for the "area+length+corners" statistics of grand ensembles of 2D directed paths. Using the combinatorial description, we obtain an explicit expression of the generating function for $q$-Narayana numbers which amounts to the new critical behavior in the ensemble of $T_{n,n+1}$ torus knots and in the ensemble of instantons in 5D SQED. Depending on the number of the nontrivial fugacities, we get either the critical point, or cascade of critical lines and critical surfaces. In the 5D gauge theory the phase transition is of the 3rd order, while in the ensemble of paths and ensemble of knots it is typically of the 1st order. We also discuss the relation with the integrable models.