Instanton Approach to LargeNHarish-Chandra-Itzykson-Zuber Integrals

Abstract

We reconsider the large N asymptotics of Harish-Chandra-Itzykson-Zuber integrals. We provide, using Dyson's Brownian motion and the method of instantons, an alternative, transparent derivation of the Matytsin formalism for the unitary case. Our method is easily generalized to the orthogonal and symplectic ensembles. We obtain an explicit solution of Matytsin's equations in the case of Wigner matrices, as well as a general expansion method in the dilute limit, when the spectrum of eigenvalues spreads over very wide regions.

Hydrodynamical spectral evolution for random matrices

“…In this limit, the first term on the RHS is subleading in N in comparison to the second and the time derivative on LHS. Ignoring this term, we obtain an equation for Sτ in the Hamilton-Jacobi form, ∂τ Sτ + dp ρ(p) 1 2 10) where the position variable is ρ(p) and the conjugate momentum reads ∆(p) = δSτ δρ(p) . This allows Sτ to be interpreted as an action evaluated on a physical trajectory between ρ(p;τ = 0) and ρ(p;τ ).…”

“…Later works looked at the same task from both mathematical [21] and physical point of view [10]. In this section we comment on this standard result and afterwards use analogous working to compute an expansion for the Berezin-Karpelevich type integrals [8,20] arising in the chiral Gaussian ensembles.…”

“…Now the main difficulty lies in finding a physical path joining initial ρ i (p) and final ρ f (p) spectral densities and calculating the corresponding action Sτ , which is specified by (5.16). The former problem has been solved from our workings in Sections 1 and 2 for the initial condiitions (1.6) and (2.25), while the evaluation of Sτ in the first of these is given in [10]. For a discussion of analyticity properties of (6.4) see [19].…”

“…The advantage of viewing (1.5) as a consequence of (1.9) is that a hydrodynamical description applies equally well to the case that X and X (0) have the chiral structure 10) and similarly X (0) with Z replaced by Z (0) , where Z is a standard real or complex Gaussian matrix, and with Z (0) non-random or random independent of Z. The class of random matrices (1.10) is fundamental to random matrix theory (see e..g. [17, §3.1]).…”

A New Proof of Harish-Chandra’s Integral Formula

“…The main purpose of our proof is to provide insight into the asymptotics of (1) as the rank of G increases. Leading-order asymptotics for (2) as N → ∞ were formally computed by Matytsin [6] and rigorously justified by Guionnet and Zeitouni [7,8], while Bun et al have reformulated these results in terms of particle trajectories concentrating around an instanton [9].…”