# Asymptotic expansion of a partition function related to the sinh-model

Source: [Asymptotic expansion of a partition function related to the sinh-model](<https://www.sota2.com/research/paper/68fa44b11e209131ce9130af>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T04:55:59.097Z

Authors: G. Borot, A. Guionnet, K. K. Kozlowski

Published: 2014-12-24T17:46:10.000Z

## Abstract

This paper develops a method to carry out the large-$N$ asymptotic analysis of a class of $N$-dimensional integrals arising in the context of the so-called quantum separation of variables method. We push further ideas developed in the context of random matrices of size $N$, but in the present problem, two scales $1/N^{\\alpha}$ and $1/N$ naturally occur. In our case, the equilibrium measure is $N^{\\alpha}$-dependent and characterised by means of the solution to a $2\\times 2$ Riemann--Hilbert problem, whose large-$N$ behavior is analysed in detail. Combining these results with techniques of concentration of measures and an asymptotic analysis of the Schwinger-Dyson equations at the distributional level, we obtain the large-$N$ behavior of the free energy explicitly up to $o(1)$. The use of distributional Schwinger-Dyson is a novelty that allows us treating sufficiently differentiable interactions and the mixing of scales $1/N^{\\alpha}$ and $1/N$, thus waiving the analyticity assumptions often used in random matrix theory.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1412.7721>)

[PDF](<https://arxiv.org/pdf/1412.7721.pdf>)