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 even though it is an outdated topic, it really is always constructing into new parts of physics and arithmetic. It constitutes now part of the final tradition of a theoretical physicist. Mathematical equipment encouraged through random matrix conception develop into extra robust, subtle and revel in speedily growing to be functions in physics. contemporary examples contain the calculation of common correlations within the mesoscopic approach, new purposes in disordered and quantum chaotic platforms, in combinatorial and progress versions, in addition to the hot leap forward, because of the matrix types, in dimensional gravity and string idea and the non-abelian gauge theories. The publication contains the lectures of the prime experts and covers particularly systematically a lot of those themes. it may be helpful to the experts in quite a few topics utilizing random matrices, from PhD scholars to proven scientists.

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E. deﬁned on random surfaces [2]. Similarly we will write continuum correlation functions of the geodesic distance on the corresponding random surfaces. 2 A brief history Planar graphs ﬁrst arose in combinatorics, in the groundbreaking works of Tutte [3] in the 60’s, who was able to compute generating functions for many classes of such objects, usually called maps by combinatorists. Higher genus was not considered then, and came up only later in physics works. The intrusion of matrix models in this subject occurred with the fundamental observation, due to t’Hooft [4] in the 70’s, that planar graphs appearing in QCD with a large number of colors could be viewed as Feynman diagrams for matrix models, and that moreover the size of the matrices could serve as an expansion parameter to keep track of the topology of these diagrams.

B. Conrey, More than 2/5 of the zeros of the Riemann zeta function are on the critical line, J. Reine. Ang. Math. 399, 1–26 (1989). B. W. Farmer, Mean values of L-functions and symmetry, Int. Math. Res. Notices 17, 883–908, 2000. B. W. P. O. C. Snaith, Autocorrelation of random matrix polynomials, Commun. Math. Phys. 237: 365–395, 2003. B. W. P. O. C. Snaith, Integral moments of L-functions, Proc. Lond. Math. Soc, to appear. B. Conrey and A. Ghosh, On mean values of the zeta-function, iii, Proceedings of the Amalﬁ Conference on Analytic Number Theory, Universit„ a di Salerno, 1992.

These would appear to support the dependence on D 3/4 , but do not cover a large enough range to determine the power of log D. 5e+08 Figure 5. s. of (87) divided by D3/4 (log D)−5/8 . The calculations include only twists with d < 0, d prime, and cases with even functional equation. While the picture is reasonably ﬂat, log(D) is almost constant for most of the interval in question. The ﬂatness observed therefore reﬂects the main dependence on D3/4 . ) Data in support of the second conjecture are listed in Table 2 and are plotted in Figure 6.