[Paper Review] Asymptotic behavior of the multiplicative counterpart of the Harish-Chandra integral and the $S$-transform

Pierre Mergny, Marc Potters|arXiv (Cornell University)|Jul 18, 2020

Random Matrices and Applications27 references9 citations

TL;DR

This paper establishes the asymptotic behavior of the multiplicative spherical integral—defined as the analytical extension of Jack symmetric polynomials—for general β > 0, showing it converges to the logarithm of the S-transform of the limiting spectral measure. Using saddle-point analysis, the authors derive a multiplicative counterpart to the Parisi-Guionnet-Maïda theorem for the HCIZ integral, where the S-transform linearizes free multiplicative convolution in the large-N limit.

ABSTRACT

In this note, we study the asymptotic of spherical integrals, which are analytical extension in index of the normalized Schur polynomials for $β=2$ , and of Jack symmetric polynomials otherwise. Such integrals are the multiplicative counterparts of the Harish-Chandra-Itzykson-Zuber (HCIZ) integrals, whose asymptotic are given by the so-called $R$-transform when one of the matrix is of rank one. We argue by a saddle-point analysis that a similar result holds for all $β>0$ in the multiplicative case, where the asymptotic is governed by the logarithm of the $S$-transform. As a consequence of this result one can calculate the asymptotic behavior of complete homogeneous symmetric polynomials.

Motivation & Objective

• To extend the asymptotic analysis of Harish-Chandra-Itzykson-Zuber (HCIZ) integrals to their multiplicative counterparts for general β > 0.
• To establish a connection between the asymptotic behavior of Jack symmetric polynomials and the S-transform in the large-N limit.
• To generalize the Parisi-Guionnet-Maïda result from additive to multiplicative free probability via spherical integrals.
• To derive the asymptotic behavior of complete homogeneous symmetric polynomials using the S-transform framework.

Proposed method

• The authors use saddle-point analysis to study the asymptotic behavior of the multiplicative spherical integral, defined as the analytical extension of Jack polynomials.
• They relate the spherical integral to the Heckman-Opdam hypergeometric function, which generalizes Schur and zonal polynomials for β = 2 and β = 1,4.
• The key equation is the asymptotic identity: lim_{N→∞} (2/(Nβ)) ln[j_{⌊Nβz/2⌋}^{(2/β)}(a)/j_{⌊Nβz/2⌋}^{(2/β)}(1,…,1)] = H^S_μ_A(z), where H^S is the S-transform logarithm.
• The S-transform is defined via the modified S-transform Ŝ_μ(z) = z/(z+1) * T_μ^{(-1)}(z), with T_μ(z) = zG_μ(z) - 1.
• The analysis assumes the empirical spectral measure of the matrix A_N converges to a deterministic measure μ_A, and the vector z is scaled such that only one entry is non-zero in the limit.
• The method relies on the invariance of the integral under permutation of eigenvalues and the normalization I^{(β)}_0(z) = 1.

Experimental results

Research questions

• RQ1 How does the multiplicative spherical integral behave asymptotically for general β > 0 in the large-N limit?
• RQ2 Can the S-transform be derived as the asymptotic generator of the multiplicative spherical integral, analogous to the R-transform in the additive case?
• RQ3 What is the limiting behavior of complete homogeneous symmetric polynomials in terms of the S-transform?
• RQ4 Does the saddle-point method yield a universal asymptotic formula for Jack polynomials indexed by a single large part?
• RQ5 Under what conditions does the multiplicative spherical integral converge to the logarithm of the S-transform?

Key findings

• The asymptotic behavior of the multiplicative spherical integral is governed by the logarithm of the S-transform: lim_{N→∞} (2/(Nβ)) ln[j_{⌊Nβz/2⌋}^{(2/β)}(a)/j_{⌊Nβz/2⌋}^{(2/β)}(1,…,1)] = H^S_μ_A(z).
• For β = 2, the Jack polynomials reduce to Schur polynomials, and the normalized Schur polynomial with index (k,0,…,0) converges to the complete homogeneous symmetric polynomial h_k(a), whose logarithmic asymptotic is given by H^S_μ_A(z).
• When the spectral measure μ_A is uniform on [0,2], the asymptotic function H^S_μ_A(z) is explicitly computed as z(ln(2z/|z+1+W(-(z+1)e^{-(z+1)}))| - 1) - ln|W(-(z+1)e^{-(z+1)})|, where W is the Lambert W function.
• Numerical results confirm convergence of (1/N)ln(h_k(a)/h_k(1,…,1)) to H^S_μ_A(z) for equidistributed a_i in [0,2], with convergence improving as N increases.
• The result implies that low-rank Schur polynomials (with few non-zero parts) asymptotically factorize into products of complete homogeneous symmetric polynomials.
• The framework suggests that in the full-rank regime, where both vectors a and z converge to full measures, the Heckman-Opdam function may admit a similar asymptotic expansion, though this remains an open question.