No abstract

No abstract

Let $N(L)$ be the number of eigenvalues, in an interval of length $L$, of a matrix chosen at random from the Gaussian Orthogonal, Unitary or Symplectic ensembles of ${\cal N}$ by ${\cal N}$ matrices, in the limit ${\cal N}\rightarrow\infty$. We prove that $[N(L) - \langle N(L)\rangle]/\sqrt{\log L}$ has a Gaussian distribution when $L\rightarrow\infty$. This theorem, which requires control of all the higher moments of the distribution, elucidates numerical and exact results on chaotic quantum systems and on the statistics of zeros of the Riemann zeta function. \noindent PACS nos. 05.45.+b, 03.65.-wComment: 13 page

Inventiones mathematicae manuscript No.

We show that the tritronquée solution yt of the Painlevé equation P I that behaves algebraically for large z with arg z = π/5, is analytic in a region containing the sector z = 0, arg z ∈ − 3π 5 , π and the disk z : |z| < 37 20 . This implies the Dubrovin conjecture, an important open problem in the theory of Painlevé transcendents. The method, building on a technique developed in [4], is general and constructive. As a byproduct, we obtain the value of the tritronquée and its derivative at zero, also important in applications, within less than 1/100 rigorous error bounds. arXiv:1209.1009v2 [math.CA] 23 Oct 2014 24 O. COSTIN 1 , M. HUANG 2 , S. TANVEER 1 8.1. Values of intermediate constants for ρ = 3. The numerical values of these constants might be helpful to the reader who would like to double-check the estimates. These are: J M = 0.282580···, jm = 0.64374···, Y 1,M = 1.16314···, Y 1,R,M = 0.132618···, E M = 0.0490292··· z 2,R,M = 0.54226···, z 2,M = 0.91863···, Mq = 0.066702···, M L,q = 0.075708···, V M = 0.2239···, T M = 0.0385··· M 1 = 1.13838···, M 2 = 0.04303···, M 3 = 0.28346···, M 4 = 0.45227···, M 5 = 0.05430···, M 6 = 0.00231···, M 7 = 0.02018··· 9. Acknowledgments