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

Random matrices are broadly 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 even though it is an outdated topic, it's continuously constructing into new parts of physics and arithmetic.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 tools encouraged via random matrix idea turn into extra robust, subtle and revel in swiftly starting to be purposes in physics. contemporary examples comprise the calculation of common correlations within the mesoscopic procedure, new purposes in disordered and quantum chaotic structures, in combinatorial and development types, in addition to the new step forward, as a result of the matrix versions, in dimensional gravity and string conception and the non-abelian gauge theories. The ebook involves the lectures of the top experts and covers fairly systematically a lot of those issues. it may be priceless to the experts in a variety of matters utilizing random matrices, from PhD scholars to proven scientists.

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The result is that [23] ˜ |Z(U, 0)|s dµ(U ) = G2 Γ(3s + 7)Γ(2s + 3) . Γ(2s + 6)Γ(s + 4)Γ(s + 3)Γ(s + 2) (104) We note in this context that Katz [19] has found a one-parameter family of L-functions over a ﬁnite ﬁeld whose value distribution in the limit as the size of the ﬁeld grows is related to G2 . Thus the random matrix moments (104) determine the value distribution of these L-functions. The random matrix calculations extend straightforwardly to all of the exceptional Lie groups. It would be very interesting indeed to know whether the others also describe families of L-functions over ﬁnite ﬁelds.

A 4-valent planar graph with hard dimers, represented by thickened edges. The corresponding graph obtained by shrinking the dimers (b) has both 4-valent and 6-valent vertices. The correspondence is three-to-one per dimer, as shown. decompositions into two 4-valent vertices and one dimer). This is the simplest instance of matter coupled to 2D quantum gravity we could think of, and it indeed corresponds to graphs with speciﬁc valence weights. Going back to the purely mathematical interpretation of (19), we start to feel how simple matrix integrals can be used as tools for generating all sorts of graphs whose duals tessellate surfaces of arbitrary given topology.

89706709 28 APPLICATIONS OF RANDOM MATRICES IN PHYSICS and (1 − ezq −z )−1 G(z1 , . . f. ) Note that G has simple poles when zi = zj , i = j. An evaluation of the contour integral in terms of residues conﬁrms the identity by giving (91). 2 (2πi)2k ··· × 1 T T Wk (log 0 t 2π )(1 1 + O(t− 2 + ))dt, (95) x Pk e2 j=1 zj −zj+k ˜ 1 , . . , z2k )∆2 (z1 , . . , z2k ) G(z 2k 2k i=1 zi dz1 . . dz2k , (96) the path of integration being the same as in (92), and k k ˜ 1 , . . , z2k ) = Ak (z1 , . . , z2k ) G(z ζ(1 + zi − zj+k ), (97) i=1 j=1 with k k 1− Ak (z) = p i=1 j=1 × 1 k 0 j=1 1 p1+zi −zj+k 1− e(θ) p1/2+zj −1 1− e(−θ) p1/2−zj+k −1 dθ (98) ˜ has the same pole structure as G.