Random-matrix theory is used to show that the proximity to a superconductor opens a gap in the excitation spectrum of an electron gas confined to a billiard with a chaotic classical dynamics. In contrast, a gapless spectrum is obtained for a non-chaotic rectangular billiard, and it is argued that this is generic for integrable systems. preprint: cond-mat/9604058

We calculate the probability distnbution of the matnx Q = -ihS 1 3S/3E for a chaotic System with scattenng matnx S at energy E The eigenvalues τ, of Q are the so-called proper delay times, mtroduced by Wigner and Smith to descnbe the time dependence of a scattenng process The distnbution of the inverse delay times turns out to be given by the Laguerre ensemble from random matnx theory [S0031 -9007(97) [8][9][10] evidence that an ensemble of chaotic bilhards contammg a small openmg (through which N modes can propagate at energy E) has a uniform distnbution of S in the group of N X N umtary matrices-restncted only by fundamental symmetnes This universal distnbution is the circular ensemble of random-matnx theory [11], mtroduced by Dyson for its mathematical simphcity [12] The eigenvalues e"^ of S m the circular ensemble are distributed accordmg to The solution of this 40 year old problem is presented here We have found that the eigenvalues of Q are independent of S [26] The distnbution of the inverse delay times y" = l/r" turns out to be the Laguerre ensemble of random-matnx theory,(2) but with an unusual ,/V-dependent exponent (The function P is zero if any one of the T"'S is negative ) The correlation functions of the T"'S consist of senes over (generahzed) Laguene polynomials [27], hence the name "Laguerre ensemble " The eigenvectors of Q are not independent of S, unless β = 2 (which is the case of broken time-reversal symmetry) However, for any β the conelations can be transformed away if we replace Q by the symmetrized matnx (3) which has the same eigenvalues äs Q The matnx of eigenvectors U which diagonahzes QE = U X diag(ri, ,r N )U^ is independent of S and the T"'S, and umformly distributed m the orthogonal, umtary, or symplectic group (for β = l, 2, or 4, respectively) The distnbution (2) confirms the conjecture by Fyodorov and Sommers [19] that the distnbution of tr Q has an algebraic Although the time-delay matnx was interpreted by Smith äs a representation of the "time operator," this Interpretation is ambiguous [19] The ambiguity anses because a wave packet has no well-defmed energy There is no ambiguity m the apphcation of Q to transport Problems where the mcommg wave can be regarded monochromatic, hke the low-frequency response of a chaotic cavity [21,22,28]

In the past decade modern societies have developed enormous communication and social networks. Their classification and information retrieval processing has become a formidable task for the society. Because of the rapid growth of the World Wide Web, and social and communication networks, new mathematical methods have been invented to characterize the properties of these networks in a more detailed and precise way. Various search engines extensively use such methods. It is highly important to develop new tools to classify and rank a massive amount of network information in a way that is adapted to internal network structures and characteristics. This review describes the Google matrix analysis of directed complex networks demonstrating its efficiency using various examples including the World Wide Web, Wikipedia, software architectures, world trade, social and citation networks, brain neural networks, DNA sequences, and Ulam networks. The analytical and numerical matrix methods used in this analysis originate from the fields of Markov chains, quantum chaos, and random matrix theory.