Free Energies and Fluctuations for the Unitary Brownian Motion

Abstract

Abstract:We show that the Laplace transforms of traces of words in independent unitary Brownian motions converge towards an analytic function on a non trivial disc. These results allow one to study the asymptotic behavior of Wilson loops under the unitary Yang-Mills measure on the plane with a potential. The limiting objects obtained are shown to be characterized by equations analogue to Schwinger-Dyson's ones, named here after Makeenko and Migdal.

Yang–Mills Measure and the Master Field on the Sphere

“…The following lemma is a reformulation of a lemma of Lévy [41,Lemma 6.28]. See also Dahlqvist [13,Lemma 21]. We give a slightly different proof, relying on properties of the winding number in place of a dimension-counting argument.…”

“…There is a system of relations, discovered by Makeenko and Migdal [45], indexed by families of embedded loops, between the expectations under the Yang-Mills measure of polynomials in the traces of loop holonomies. These have now been proved for the whole plane by Lévy [40] and Dahlqvist [13] and for any compact surface by Driver, Gabriel, Hall and Kemp [18]. They belong to the class of Schwinger-Dyson equations, a family of equations obtained by generalizing integration-by-parts formulas to the setting of functional integrals.…”

“…The interest of such a set-up from the viewpoint of random matrix theory was first raised in the mathematics literature by Singer [50], who made several conjectures, based on earlier work in physics [26,27,34,35]. The high-dimensional limit of the Yang-Mills measure when Σ is the whole plane has since been studied by Xu [54], Sengupta [49], Lévy [40], Anshelevich and Sengupta [1], Dahlqvist [13] and others [8,24]. We focus here on the case where the surface Σ is a sphere.…”

“…In the whole plane case, moment estimates for unitary Brownian motion provide the needed concentration, and the Makeenko-Migdal equations may be augmented by a further equation, such that the whole system of equations then characterizes the limit field. So the programme has been completed in that case [13,40]. However, as noted in [18], the concentration and characterization problems have remained open in general.…”

The generalized master fields

“…One can tackle the problem of fluctuations around the limit or of moderate deviations with the same strategy. In [8], it has been shown that the approach of [17], can be generalized for the Brownian motion, to get convergence results for Laplace transforms of the considered random variables, yielding local central limit theorems. For Lévy processes, this remains an open question.…”