First Order Asymptotics of Matrix Integrals; A Rigorous Approach Towards the Understanding of Matrix Models

By Alice Guionnet. 2002

71 citations recorded by the source corpus.

We investigate the large N limit of spectral measures of matrices which relate to the Gibbs measures of a number of statistical mechanical systems on random graphs. These include the Ising and Potts models on random graphs. For most of these models, we prove that the spectral measures converge almost surely and describe their limit via solutions to an Euler equation for isentropic flow with negative pressure p(ρ)=−3−1π2ρ3.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/1e8714af434f4d5e495fba72999ffa80ef7fe1de

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Citations in the corpus

Asymptotics of unitary and orthogonal matrix integrals (2006, methodology): “…to [18], by where the inﬁmum is taken over m, ρ so that the measure-valued process µ t ( dx ) = ρ t ( x ) dx is a continuous process, µ 0 = µ A , µ 1 = µ B and The inf over ( ρ t , m t ) is taken (see [14]) at the solution of an Euler equation for isentropic ﬂow with negative pressure − π 2 3 ρ 3 .”
A Fourier view on the R-transform and related asymptotics of spherical integrals (2005, background): “They have been then extensively studied in the framework of so-called matrix models that are related to the problem of enumerating maps (after [16], it has been developed in physics for example in [27], [19] or [21], in mathematics in [6] or [11]; a very nice introduction to these links is provided ”
Limit shapes and the complex Burgers equation (2005, unknown): “Its rigorous mathematical study appears in the work of A. Guionnet [ 5 ].”