Asymptotics of the Harish-Chandra-Itzykson-Zuber (HCIZ) integral

by Erik Aurell (KTH)

Thursday 6 Nov 2025, 13:00 → 15:00 Europe/Stockholm

Albano 3: 5230 - Xenon (12 seats) (Albano Building 3)

In the course of a follow-up of a recent paper [1] we came upon an integral over the orthogonal group O(2N): I(A,B) = \int dO exp[ i Tr (A O B O^T )] where A and B are 2-by-2 block diagonal matrices, and all the blocks are proportional to (0 , 1 \\ -1 , 0). This is an example of a Harish-Chandra-Itzykson-Zuber (HCIZ) integral [2], commonly written, for Hermitian matrices A and B, a real or complex parameter t, and integrals over the unitary group dU as I(A,B) = \int dU exp[ t Tr (A U B U^{\dagger} )]. In our application t=i and the dimensionality N is extremely high, say about $10^{80}$.  

I will review the asymptotics of the HCIZ integral in high dimensions, following mostly [3]. For real parameter t it can be mapped on the probability of the eigenvalues of A as "1D particle positions" changing into the eigenvalues of B as other "1D particle positions", under an over-damped dynamics with long-range interactions. I will also say something of other applications of HCIZ integral to the interface of statistical physics and AI, see e.g. [4].

[1] Aurell, Hackl & Kieburg, Quantum Sci. Technol. 10 045068 (2025) 'Average entanglement entropy of a small subsystem in a pure Gaussian state ensemble'

[2] Terence Tao's blog, February 8 (2013) 'The Harish-Chandra-Itzykson-Zuber integral formula'

[3] Bun, Bouchaud, Majumdar & Potters, Phys. Rev. Lett. 113, 070201 (2014) 'Instanton approach to large N Harish-Chandra-Itzykson-Zuber integrals' [arXiv:1403:7763] 

[4] Barbier, Camilli, Nguyen, Pastore & Skerk , arXiv:2510.24616 (2025) 'Statistical physics of deep learning: Optimal learning of a multi-layer perceptron near interpolation'