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...and the so called Schwinger–Dyson (or master-loop) equations. Such equations were already the key to obtain fine estimates on the Stieltjes transform of Gaussian generalized band matrices in [14]. In =-=[4]-=-, they were used to study the asymptotics of matrix models on the unitary group. Our approach combines ideas of [14] to estimate Stieltjes transform and the necessary adaptations to unitary matrices a...

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...opological interpretation of this expansion, thereby placing the HCIZ model on similar footing with the more developed theory of topological expansion in Hermitian matrix models [1, 2, 3, 9, 21]. See =-=[6, 7]-=- for previous results in this direction. In this article, we give a thorough combinatorial analysis of the monotone single Hurwitz numbers ~Hg(α) = ~Hg(α, (1 d)), which count branched covers as above ...

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...otic behaviour of the trace of polynomial functions of the m-tuple UN and the deterministic contractions ρN (bi). 1.2. Initial value problem with potential. 1.2.1. Collins, Guionnet and Maurel-Segala =-=[5]-=- considered a noncommutative initial value problem which generalizes (1.1), namely (1.2) τ ⊗ τ(∂ip) + τ((DiV )p) = 0 τ |B = σ } . Here V ∈ L is a fixed polynomial (the “potential”), and Di is the Laur...

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...studied power-like (Fisher-Hartwig) singularities in [13]. Moreover, the asymptotics for partition functions in multi-matrix models are studied, too. For instance, one may refer to a series of papers =-=[4, 11, 12]-=- done by Guionnet and her colleagues. In this work, we plan to derive the asymptotic expansion of logZN(t) as N → ∞. Based on (1.10), it is sufficient to derive the asymptotics of the one-point correl...

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...he empirical measures {LNk }1≤k≤d converge almost surely under PN,aVβ towards probability measures {µaVk }1≤k≤d on the real line. In the case r = 1 this result is already a consequence of [GMS06] and =-=[CGMS09]-=-. The existence and study of the equilibrium measures is performed in Section 3. From the representation of the density given in Theorem 2.2 (see Section 4), we deduce the following approximate transp...

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...mutative polynomial plays the role of the potential of a Gibbs measure, the normalized logarithm of Laplace tranforms is called the free energy and have been studied in several places, for example in =-=[20, 11, 7]-=-. In the pioneering work [8], formal expansions have been proposed for several physical models. In [10, 12], technics have been developed to study formal expansions for model of random matrices with p...

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...ed. More results here were obtained by Collins and Matsumoto in [10], and Zinn-Justin [22]. These combinatorial methods have a number of concrete applications, notably to random matrix questions [6], =-=[9]-=-, [11], [12], [16], [18], [20], and to invariance questions [5]. The present paper is a continuation of our previous work [3]. We will use the “elementary expansion” approach to the Weingarten functio...