One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References.

Quantum Inverse Scattering Method and Correlation Functions

We study the characteristic polynomials Z(U, θ) of matrices U in the Circular Unitary Ensemble (CUE) of Random Matrix Theory. Exact expressions for any matrix size N are derived for the moments of |Z| and Z/Z*, and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log Z as N→∞. In the limit, we show that these two distributions are independent and Gaussian. Costin and Lebowitz [15] previously found the Gaussian limit distribution for Im log Z using a different approach, and our result for the cumulants proves a conjecture made by them in this case. We also calculate the leading order N→∞ asymptotics of the moments of |Z| and Z/Z*. These CUE results are then compared with what is known about the Riemann zeta function ζ (s) on its critical line Re s= 1/2, assuming the Riemann hypothesis. Equating the mean density of the non-trivial zeros of the zeta function at a height T up the critical line with the mean density of the matrix eigenvalues gives a connection between N and T. Invoking this connection, our CUE results coincide with a theorem of Selberg for the value distribution of log ζ(1/2+iT) in the limit T→∞. They are also in close agreement with numerical data computed by Odlyzko [29] for large but finite T. This leads us to a conjecture for the moments of |ζ(1/2+it) |. Finally, we generalize our random matrix results to the Circular Orthogonal (COE) and Circular Symplectic (CSE) Ensembles.

/pdf/random-matrix-theory-and-z-1-2-it-3xihgkn7e2.pdf

Random Matrix Theory and ζ(1/2+it)

We review selected advances in the theoretical understanding of complex quantum many-body systems with regard to emergent notions of quantum statistical mechanics. We cover topics such as equilibration and thermalisation in pure state statistical mechanics, the eigenstate thermalisation hypothesis, the equivalence of ensembles, non-equilibration dynamics following global and local quenches as well as ramps. We also address initial state independence, absence of thermalisation, and many-body localisation. We elucidate the role played by key concepts for these phenomena, such as Lieb-Robinson bounds, entanglement growth, typicality arguments, quantum maximum entropy principles and the generalised Gibbs ensembles, and quantum (non-)integrability. We put emphasis on rigorous approaches and present the most important results in a unified language.

https://iopscience.iop.org/article/10.1088/0034-4885/79/5/056001/pdf

Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems.

Fundamental theory: Holomorphic functions and domains of holomorphy Implicit functions and analytic sets The Poincare, Cousin, and Runge problems Pseudoconvex domains and pseudoconcave sets Holomorphic mappings Theory of analytic spaces: Ramified domains Analytic sets and holomorphic functions Analytic spaces Normal pseudoconvex spaces Bibliography Index.

Function theory in several complex variables

Quantum inverse scattering method and correlation functions

Recent results of Katz and Sarnak [8, 9] suggest that the low-lying zeros of families of L-functions display the statistics of the eigenvalues of one of the compact groups of matrices U(N), O(N) or USp(2N). We here explore the link between the value distributions of the L-functions within these families at the central point s= 1/2 and those of the characteristic polynomials Z(U,θ) of matrices U with respect to averages over SO(2N) and USp(2N) at the corresponding point θ= 0, using techniques previously developed for U(N) in [10]. For any matrix size N we find exact expressions for the moments of Z(U,0) for each ensemble, and hence calculate the asymptotic (large N) value distributions for Z(U,0) and log Z(U,0). The asymptotic results for the integer moments agree precisely with the few corresponding values known for L-functions. The value distributions suggest consequences for the non-vanishing of L-functions at the central point.

http://www.hpl.hp.com/techreports/2000/HPL-BRIMS-2000-05.pdf

Random matrix theory and L-functions at s=1/2

We give a new heuristic for all of the main terms in the integral moments of various families of primitive $L$-functions. The results agree with previous conjectures for the leading order terms. Our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary, orthogonal, or symplectic matrices, where the moments are defined by the appropriate group averages. This lends support to the idea that arithmetical $L$-functions have a spectral interpretation, and that their value distributions can be modelled using Random Matrix Theory. Numerical examples show good agreement with our conjectures.

https://web.williams.edu/Mathematics/sjmiller/public_html/ntandrmt/handouts/conrey/ConreyFarmerKeatingRubinsteinSnaith_IntegralMomentsLFns.pdf

Integral moments of L-functions

We give a new heuristic for all of the main terms in the integral moments of various families of primitive L-functions. The results agree with previous conjectures for the leading order terms. Our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary, orthogonal, or symplectic matrices, where the moments are defined by the appropriate group averages. This lends support to the idea that arithmetical L-functions have a spectral interpretation, and that their value distributions can be modeled using Random Matrix Theory. Numerical examples show good agreement with our conjectures.

In upcoming papers by Conrey, Farmer and Zirnbauer there appear conjectural formulas for averages, over a family, of ratios of products of shifted L-functions. In this paper we will present various applications of these ratios conjectures to a wide variety of problems that are of interest in number theory, such as lower order terms in the zero statistics of L-functions, mollified moments of L-functions and discrete averages over zeros of the Riemann zeta function. In particular, using the ratios conjectures we easily derive the answers to a number of notoriously difficult computations.

https://people.maths.bris.ac.uk/~mancs/papers/new_ratiosI.pdf

Nina C Snaith

Papers

Random Matrix Theory and ζ(1/2+it)

Random matrix theory and L-functions at s=1/2

Integral moments of L-functions

Integral moments of L-functions

Applications of the L-functions ratios conjectures