By Gaëtan Borot, Alice Guionnet, Karol K. Kozlowski

ISBN-10: 331933378X

ISBN-13: 9783319333786

ISBN-10: 3319333798

ISBN-13: 9783319333793

This publication elaborates at the asymptotic behaviour, while N is huge, of sure N-dimensional integrals which generally ensue in random matrices, or in 1+1 dimensional quantum integrable versions solvable through the quantum separation of variables. The advent provides the underpinning motivations for this challenge, a old evaluation, and a precis of the tactic, that is appropriate in larger generality. The center goals at proving a ramification as much as o(1) for the logarithm of the partition functionality of the sinh-model. this is often accomplished by means of a mixture of power thought and big deviation idea as a way to grab the major asymptotics defined through an equilibrium degree, the Riemann-Hilbert method of truncated Wiener-Hopf which will examine the equilibrium degree, the Schwinger-Dyson equations and the boostrap way to eventually receive a selection of correlation services and the single 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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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 .

We will outline the main ideas of the method on the example of the open quantum Toda chain Hamiltonian with (N + 1)-particles [117]: N+1 HTd = a=1 p2a + exN+1 −x1 + 2 N exa −xa+1 . 1) a=1 Above, xa is to be understood as the operator of multiplication by the ath coordinate xa · (x) = xa (x) while pa is the canonically conjugated operator, pa · (x) = −i ∂ (x)/∂xa , so that xa , pb = δa,b i . Here, x denotes a N + 1 dimensional vector x = (x1 , . . , xN+1 ). 5 The Integrals Issued from the Method of Quantum Separation of Variables 33 xn and pn , the Toda chain Hamiltonian is a multi-dimensional partial differential operator acting on the Hilbert space HToda = L 2 RN+1 , dN+1 x .

28) a=1 This integral is related to zN [W ] by a rescaling of the integration variables. The exponent α is fixed by the growth of the original potential W at infinity. Finally, the potential V should depend on N and correspond to some rescaling of the original potential W . 27) β should be compared to the parameter in the β ensembles. So, the special case β = 1 is put in 2 parallel with the special case of β = 2-ensembles. 5 The Integrals Issued from the Method of Quantum Separation of Variables 41 large-N asymptotic expansion of the rescaled partition function ZN [V ] and this in the case where • the potential V is smooth, strictly convex, has sub-exponential growth and is Nindependent; • 0 < α < 1/6; The first assumption is more than enough to carry the large deviation analysis, which gives the leading order of ln ZN [V ], while the second assumption appears in the course of the bootstrap analysis of the Schwinger–Dyson equations.

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Asymptotic Expansion of a Partition Function Related to the Sinh-model by Gaëtan Borot, Alice Guionnet, Karol K. Kozlowski

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