arXiv subjects

Jonathan P. Keating

Publications and source records attributed to Jonathan P. Keating.

41 records · Page 3Linked to original sources

arXiv2013

Freezing Transitions and Extreme Values: Random Matrix Theory, $ζ(1/2+it)$, and Disordered Landscapes

We argue that the freezing transition scenario, previously conjectured to occur in the statistical mechanics of 1/f-noise random energy models, governs, after reinterpretation, the value distribution of the maximum of the modulus of the characteristic polynomials p_N(θ) of large N\times N random unitary (CUE) matrices; i.e. the extreme value statistics of p_N(θ) when N \rightarrow\infty. In addition, we argue that it leads to multifractal-like behaviour in the total length μ_N(x) of the intervals in which |p_N(θ)|>N^x, x>0, in the same limit. We speculate that our results extend to the large values taken by the Riemann zeta-function ζ(s) over stretches of the critical line s=1/2+it of given constant length, and present the results of numerical computations of the large values of ζ(1/2+it). Our main purpose is to draw attention to the unexpected connections between these different extreme value problems.

math-ph↗

arXiv2013

n-particle quantum statistics on graphs

We develop a full characterization of abelian quantum statistics on graphs. We explain how the number of anyon phases is related to connectivity. For 2-connected graphs the independence of quantum statistics with respect to the number of particles is proven. For non-planar 3-connected graphs we identify bosons and fermions as the only possible statistics, whereas for planar 3-connected graphs we show that one anyon phase exists. Our approach also yields an alternative proof of the structure theorem for the first homology group of n-particle graph configuration spaces. Finally, we determine the topological gauge potentials for 2-connected graphs.

math-ph↗

arXiv2012

Freezing Transition, Characteristic Polynomials of Random Matrices, and the Riemann Zeta-Function

We argue that the freezing transition scenario, previously explored in the statistical mechanics of 1/f-noise random energy models, also determines the value distribution of the maximum of the modulus of the characteristic polynomials of large N x N random unitary (CUE) matrices. We postulate that our results extend to the extreme values taken by the Riemann zeta-function zeta(s) over sections of the critical line s=1/2+it of constant length and present the results of numerical computations in support. Our main purpose is to draw attention to possible connections between the statistical mechanics of random energy landscapes, random matrix theory, and the theory of the Riemann zeta function.

math-ph↗

arXiv2007

Resummation and the semiclassical theory of spectral statistics

We address the question as to why, in the semiclassical limit, classically chaotic systems generically exhibit universal quantum spectral statistics coincident with those of Random Matrix Theory. To do so, we use a semiclassical resummation formalism that explicitly preserves the unitarity of the quantum time evolution by incorporating duality relations between short and long classical orbits. This allows us to obtain both the non-oscillatory and the oscillatory contributions to spectral correlation functions within a unified framework, thus overcoming a significant problem in previous approaches. In addition, our results extend beyond the universal regime to describe the system-specific approach to the semiclassical limit.

nlin.CD↗

arXiv2002

Negative moments of characteristic polynomials of random GOE matrices and singularity-dominated strong fluctuations

We calculate the negative integer moments of the (regularized) characteristic polynomials of N x N random matrices taken from the Gaussian Orthogonal Ensemble (GOE) in the limit as $N \to \infty$. The results agree nontrivially with a recent conjecture of Berry & Keating motivated by techniques developed in the theory of singularity-dominated strong fluctuations. This is the first example where nontrivial predictions obtained using these techniques have been proved.

math-ph↗