Asymptotic of Coulomb gas integral, Temperley-Lieb type algebras and pure partition functions

TL;DR

This paper analyzes the asymptotic behavior of Coulomb gas integrals and constructs pure partition functions for multiple SLE systems, proving linear independence of solutions for irrational  and explicitly computing associated matrices.

Contribution

It introduces a novel method to establish linear independence of ground state solutions and constructs pure partition functions for multiple SLE systems using Temperley-Lieb algebra techniques.

Findings

Proves linear independence of ground state solutions for irrational .

Constructs pure partition functions for multiple SLE systems.

Provides explicit expression for the meander matrix determinant.

Abstract

In this supplementary note, we study the asymptotic behavior of several types of Coulomb gas integrals and construct the pure partition functions for multiple radial $\mathrm{SLE}(\kappa)$ and general multiple chordal $\mathrm{SLE}(\kappa)$ systems. For both radial and chordal cases, we prove the linear independence of the ground state solutions $J_{\alpha}^{(m,n)}(\boldsymbol{x})$ to the null vector equations for irrational values of $\kappa \in (0,8)$. In particular, we show that the ground state solutions $J^{(m,n)}_\alpha \in B_{m,n}$, indexed by link patterns $\alpha$ with $m$ screening charges, are linearly independent when $\kappa$ is irrational. This is achieved by constructing, for each link pattern $\beta$, a dual functional $l_\beta \in B^{*}_{m,n}$ such that the meander matrix of the corresponding Temperley-Lieb type algebra is given by $M_{\alpha\beta} = l_{\beta}(J^{(m,n)}_\alpha)$. The determinant of this matrix admits an explicit expression and is nonzero for irrational $\kappa$, establishing the desired linear independence. As a consequence, we construct the pure partition functions $Z_{\alpha}(\boldsymbol{x})$ of the multiple $\mathrm{SLE}(\kappa)$ systems for each link pattern $\alpha$ by multiplying the inverse of the meander matrix. This method can also be extended to the asymptotic analysis of the excited state solutions $K_{\alpha}$ in both radial and chordal cases.