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

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.