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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.01306 |
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Table of Contents:
- 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 $\mathrm{SLE}(κ)$ and general multiple chordal $\mathrm{SLE}(κ)$ systems. For both radial and chordal cases, we prove the linear independence of the ground state solutions $J_α^{(m,n)}(\boldsymbol{x})$ to the null vector equations for irrational values of $κ\in (0,8)$. In particular, we show that the ground state solutions $J^{(m,n)}_α\in B_{m,n}$, indexed by link patterns $α$ with $m$ screening charges, are linearly independent when $κ$ is irrational. This is achieved by constructing, for each link pattern $β$, a dual functional $l_β\in B^{*}_{m,n}$ such that the meander matrix of the corresponding Temperley-Lieb type algebra is given by $M_{αβ} = l_β(J^{(m,n)}_α)$. The determinant of this matrix admits an explicit expression and is nonzero for irrational $κ$, establishing the desired linear independence. As a consequence, we construct the pure partition functions $Z_α(\boldsymbol{x})$ of the multiple $\mathrm{SLE}(κ)$ systems for each link pattern $α$ by multiplying the inverse of the meander matrix. This method can also be extended to the asymptotic analysis of the excited state solutions $K_α$ in both radial and chordal cases.