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

Random matrices are largely 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 even though it is an outdated topic, it's continuously constructing into new parts of physics and arithmetic. It constitutes now part of the overall tradition of a theoretical physicist. Mathematical equipment encouraged by means of random matrix idea develop into extra strong, refined and revel in swiftly turning out to be purposes in physics. contemporary examples contain the calculation of common correlations within the mesoscopic process, new functions in disordered and quantum chaotic platforms, in combinatorial and development types, in addition to the new leap forward, because of the matrix types, in dimensional gravity and string concept and the non-abelian gauge theories. The ebook includes the lectures of the major experts and covers relatively systematically lots of those issues. it may be necessary to the experts in quite a few matters utilizing random matrices, from PhD scholars to proven scientists.

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**Example text**

One important point to notice is that for every cn ∈ Z. I will now describe two conjectural implications that follow from combining this formula with the random-matrix model. 3 Generalization of the Sato-Tate law to half-integer weight modular forms The Sato-Tate law describes the value distribution of the Fourier coefﬁcients ap deﬁned in (72). According to the theorems of Hasse and Deligne these √ satisfy |ap | ≤ 2 p and so may be written ap √ = 2 cos θp , 0 ≤ θ ≤ π. p The question then is: how are the angles θp distributed as the prime p varies?

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. H. E. Littlewood, Some problems in "Partitio Numerorum" III: on the expression of a number as a sum of primes, Acta Mathematica 44, 1–70, 1923.

The L-function values have been normalized so that they have the same means as the random matrix value distributions. ) The key question is obviously: what use can be made of the random matrix model for the value distribution of L-functions? I will now outline some applications that are currently being explored for the L-functions associated with 23 Random Matrices and Number Theory elliptic curves. These exploit certain explicit formulae for the values at the central point s = 1. The approach is general, but for simplicity I shall describe it in the speciﬁc context of the family deﬁned in example 2.