By Gaëtan Borot, Alice Guionnet, Karol K. Kozlowski

ISBN-10: 331933378X

ISBN-13: 9783319333786

ISBN-10: 3319333798

ISBN-13: 9783319333793

This e-book elaborates at the asymptotic behaviour, while N is big, of convinced N-dimensional integrals which usually take place in random matrices, or in 1+1 dimensional quantum integrable versions solvable through the quantum separation of variables. The advent offers the underpinning motivations for this challenge, a ancient review, and a precis of the method, that is appropriate in higher generality. The center goals at proving a diffusion as much as o(1) for the logarithm of the partition functionality of the sinh-model. this can be accomplished through a mix of capability concept and big deviation idea in an effort to grab the major asymptotics defined by means of an equilibrium degree, the Riemann-Hilbert method of truncated Wiener-Hopf so one can examine the equilibrium degree, the Schwinger-Dyson equations and the boostrap technique to eventually receive a variety of correlation features and the single of the partition functionality. This publication is addressed to researchers operating in random matrices, statistical physics or integrable platforms, or attracted to contemporary advancements of asymptotic research in these fields.

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Represents a wave packet having a dispersion in ε momentum space given by f ∈ L 1 (R). Further the function (norm) (x) ε;t ˆ = N ϕy (x) · RN a=1 dμ(y ) qt (ya ) · √ N N! 5 The Integrals Issued from the Method of Quantum Separation of Variables 35 represents the “normalisable” part of the generalised eigenfunction of the operators t(λ) associated with the eigenvalues t(λ) and a total momentum ε. One speaks of a separation of variables since the normalisable part of the generalised eigenfunction is given by a product of functions in one variable qt (λa ), a = 1, .

In other words, one should find how the given operator on HToda acts on the space Hsep where the separation of variables is realised. This inverse problem has been solved for different examples of quasi-local operators and for various models [110, 113, 123–128]. • The Toda Chain The resolution of the inverse problem for the Toda chain has been pioneered by Babelon [123, 124] in 2002 and further developed in the works [125, 128]. These results, along with the unitarity of the separation of variables transform U lead to multiple integral representations for the form factors.

8). The pair (tσ , qσ ) provides ones with a solution to the Baxter T-Q equation if and only if the parameters {σl }N+1 l=1 and ζ satisfy to the quantisation conditions, for any k ∈ {1, . . 19) Yσ (τ ) 1 1 ln 1 + + σk − τ + i /2 σk − τ − i /2 ϑ (τ − i /2) ϑ (τ + i /2) σk = ε. 19) gives rise to the solution (tσ (λ), qσ (λ)) to the T-Q equations. 2 Multiple Integral Representations The objects of main interest to the physics of a quantum integrable model are its correlation functions, namely expectation values of products of certain physically relevant operators taken between two eigenstates of the Hamiltonian of the model.

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

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