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

Author:Gaëtan Borot, Alice Guionnet & Karol K. Kozlowski
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(2.5.5)

and the proof shows that the remainder is uniform in H and V provided that H is regular enough and that V satisfies to the hypothesis given in (2.2.2)–(2.2.4).

Furthermore, the expansion (2.5.5) involves

(2.5.6)

with

(2.5.7)

Note that, in (2.5.6), the indicates the variable of the function on which the operator acts. Given sufficiently regular functions H, V, we obtain in Section 6.​3 and more precisely in Proposition 6.​3.​10 the large-N asymptotic behaviour of . We then have all the elements to calculate the large-N asymptotic behaviour of the partition function . For this purpose, we observe that, when , the partition function associated to a quadratic potential can be explicitly evaluated as shown in Proposition D.0.19. One can also show (cf. Lemma D.0.18) that there exists a unique, up to a constant, quadratic potential such that its associated equilibrium measure has the same support as the one associated with V. Then is a one parameter t smooth family of strictly convex potentials, and . Furthermore, if follows from the details of the analysis that led to (2.5.5) that the remainder will be uniform in . As a consequence, by combining all of the above results and integrating equation (2.5.4) over t, we get that, in the asymptotic regime,

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