By Gaëtan Borot, Alice Guionnet, Karol K. Kozlowski
This booklet elaborates at the asymptotic behaviour, while N is huge, of definite N-dimensional integrals which usually take place in random matrices, or in 1+1 dimensional quantum integrable versions solvable by way of the quantum separation of variables. The advent offers the underpinning motivations for this challenge, a old evaluation, and a precis of the tactic, that's acceptable in better generality. The middle goals at proving a diffusion as much as o(1) for the logarithm of the partition functionality of the sinh-model. this is often completed by means of a mix of capability idea and big deviation conception on the way to grab the major asymptotics defined through an equilibrium degree, the Riemann-Hilbert method of truncated Wiener-Hopf on the way to examine the equilibrium degree, the Schwinger-Dyson equations and the boostrap solution to eventually receive a spread of correlation features and the single of the partition functionality. This booklet 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
22) for any K ≥ 0. 2. 20) of the asymptotic expansion is not valid any more: new bounded oscillatory (β) in N contributions have to be included in Fk [V ] for k ≥ 0. Heuristically speaking, this effect takes its roots in the possibility the particles have to tunnel from one cut to another [56, 57]. On the technical level, this takes its origin in the fact that the master operator K has a kernel whose dimension is given by the number of cuts minus one. For real-analytic off-critical potentials and general β > 0, the form of the all-order asymptotic expansion in the multi-cut case was conjectured in , and established in .
9) qt (ya ) · f (ε) · √ N! 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, .
Fn ] = ∂t1 =0 · · · ∂tn =0 ln PN n ti Oi exp . i=1 In fact, the cumulants are enough for computing all the n-linear statistics. Indeed, one has the reconstruction ˆ n ENV n fa (xa ) · a=1 dLN(λ) (xi ) n s = C|Ja | fk s=1 [[ 1 ; n ]]= a=1 J1 ··· Js i=1 k∈Ja . The expression for n-linear statistics involving genuine test functions in n variables belonging to the test space T (Rn ) is then obtained by density of, say, T (R) ⊗ · · · ⊗ T (R) in T (Rn ). It is advantageous to work with Cn f1 , . .
Asymptotic Expansion of a Partition Function Related to the Sinh-model by Gaëtan Borot, Alice Guionnet, Karol K. Kozlowski