By Édouard Brezin, Vladimir Kazakov, Didina Serban, Paul Wiegmann, Anton Zabrodin

Random matrices are extensively and effectively utilized in physics for nearly 60-70 years, starting with the works of Dyson and Wigner. even though it is an outdated topic, it's consistently constructing into new parts of physics and arithmetic. It constitutes now part of the final tradition of a theoretical physicist. Mathematical equipment encouraged by way of random matrix thought develop into extra strong, subtle and revel in speedily transforming into purposes in physics. contemporary examples comprise the calculation of common correlations within the mesoscopic procedure, new purposes in disordered and quantum chaotic platforms, in combinatorial and progress types, in addition to the new leap forward, as a result of matrix versions, in dimensional gravity and string idea and the non-abelian gauge theories. The publication includes the lectures of the prime experts and covers particularly systematically a lot of those subject matters. it may be worthy to the experts in numerous matters utilizing random matrices, from PhD scholars to proven scientists.

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**Sample text**

19) gives a clear geometrical meaning to the Gaussian average of our choice of f (M ): it amounts to computing the generating function for fatgraphs of given genus and 42 APPLICATIONS OF RANDOM MATRICES IN PHYSICS given vertex valencies. Such a fatgraph Γ is in turn dual to a tessellation Γ∗ of a Riemann surface of same genus, by means of ni i-valent polygonal tiles, i = 1, 2, .... The result (19) is therefore a statistical sum over discretized random surfaces (the tessellations), that can be interpreted in physical terms as the free energy of a model of discrete 2D quantum gravity.

B. P. O. C. A. , pages 301–315. nt/0012043. B. P. O. C. Snaith, Random matrix theory and the Fourier coefﬁcients of half-integral weight forms, preprint, 2004. J. Dyson, Statistical theory of the energy levels of complex systems, i, ii and iii, J. Math. Phys. 3, 140–175, 1962. C. Gutzwiller, Periodic orbits and classical quantization conditions J. Math. Phys. 12, 343–358, 1971. H. E. Littlewood, Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes, Acta Mathematica 41, 119–196, 1918.

Typically, we may take for f a monomial of the form f (M ) = (i,j)∈I Mij , I a ﬁnite set of pairs of indices. Note the presence of the normalization factor N (=the size of the matrices) in the exponential. e. real numbers) here. Like before, for a given Hermitian N × N matrix S, let us introduce the source integral Σ(S) = eT r(SM ) = e T r(S 2 ) 2N (6) easily obtained by completing the square M 2 − N (SM + M S) = (M − N S)2 − N 2 S 2 and performing the change of variable M = M − N S. We can use (6) to compute any average of the form Mij Mkl ...