Geodesic
On the Douglas-Kazakov phase transition - Weighted potential theory under constraint for probabilists
Thierry Lévy1 ; Mylène Maïda2
1 UPMC, Université Pierre et Marie Curie - LPMA - Case courrier 188 - 4, place Jussieu - 75252 Paris Cedex 05.
2 Université Lille 1 - Laboratoire Paul Painlevé - Cité Scientifique - 59655 Villeneuve d’ascq Cedex
ESAIM. Proceedings, Tome 51 (2015), pp. 89-121.

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We give a rigorous proof of the fact that a phase transition discovered by Douglas and Kazakov in 1993 in the context of two-dimensional gauge theories occurs. This phase transition can be formulated in terms of the Brownian bridge on the unitary group in large dimension. We explain how it can be understood by considering the asymptotic behaviour of the eigenvalues of the unitary Brownian bridge, and how it can be technically approached by means of Fourier analysis on the unitary group. Moreover, we advertise some more or less classical methods for solving certain minimisation problems which play a fundamental role in the study of the phase transition and more generally in what can be called ’discrete Coulomb gases’.
DOI : 10.1051/proc/201551006

Thierry Lévy 1 ; Mylène Maïda 2

1 UPMC, Université Pierre et Marie Curie - LPMA - Case courrier 188 - 4, place Jussieu - 75252 Paris Cedex 05.
2 Université Lille 1 - Laboratoire Paul Painlevé - Cité Scientifique - 59655 Villeneuve d’ascq Cedex 
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     title = {On the {Douglas-Kazakov} phase transition - {Weighted} potential theory under constraint for probabilists},
     journal = {ESAIM. Proceedings},
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Thierry Lévy; Mylène Maïda. On the Douglas-Kazakov phase transition - Weighted potential theory under constraint for probabilists. ESAIM. Proceedings, Tome 51 (2015), pp. 89-121. doi : 10.1051/proc/201551006. http://geodesic.mathdoc.fr/articles/10.1051/proc/201551006/

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