By Gaëtan Borot, Alice Guionnet, Karol K. Kozlowski

This booklet elaborates at the asymptotic behaviour, while N is huge, of yes N-dimensional integrals which generally take place in random matrices, or in 1+1 dimensional quantum integrable types solvable by means of the quantum separation of variables. The advent provides the underpinning motivations for this challenge, a ancient evaluate, and a precis of the tactic, that's appropriate in larger generality. The center goals at proving a selection as much as o(1) for the logarithm of the partition functionality of the sinh-model. this is often completed by way of a mixture of power idea and massive deviation idea for you to snatch the major asymptotics defined by way of an equilibrium degree, the Riemann-Hilbert method of truncated Wiener-Hopf with a view to examine the equilibrium degree, the Schwinger-Dyson equations and the boostrap technique to eventually receive a ramification of correlation capabilities and the only of the partition functionality. This ebook is addressed to researchers operating in random matrices, statistical physics or integrable structures, or drawn to contemporary advancements of asymptotic research in these fields.

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**Extra info for Asymptotic Expansion of a Partition Function Related to the Sinh-model**

**Example text**

Av,w corresponds to the number of edges that link the vertices v and w–. Kharchev et al. [95] introduced the N-fold integral: ˆ RN Nv v∈D a=1 (v) −N Vv (λa dλ(v) a e ) (v) λ(v) a − λb · a**
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**N × ˆ N RN a
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**2) It is well known that, then, the associated multiple integrals can be fully characterised in terms of appropriate systems of bi-orthogonal polynomials in the sense of [84]. 3) R and ˆ Qn g(λ) · f (λ) · e−N V (λ) · dλ = 0 for j ∈ {0, . . , n − 1} . j R The system of bi-orthogonal polynomials subordinate to f and g exists and is unique for instance when f and g are real-valued and monotone functions. In that case, the multiple integral of interest can be recast as a determinant which, in turn, can be evaluated in terms of the overlaps involving the polynomials Pn and Qn by carrying out linear combinations of lines and columns of the determinant: ˆ f (λ) detN j,k∈[[ 1 ; N ]] R j−1 · g(λ) k−1 ·e−N V (λ) ·dλ = N−1 ˆ n=0 R Pn f (λ) ·Qn g(λ) ·e−N V (λ) ·dλ . **