On the fluctuations of matrix elements of the quantum cat map

TL;DR

This paper investigates the statistical fluctuations of matrix elements in the quantum cat map, confirming parts of a conjecture and analyzing moments and sums of these elements in the semiclassical limit.

Contribution

It demonstrates the vanishing of the fifth centered moment in the Hecke basis and analyzes moments of sums of matrix elements in short windows, advancing understanding of quantum chaos.

Findings

Fifth centered moment vanishes in the semiclassical limit.

Third moment of sums in small windows vanishes after normalization.

Variance of sums is determined in the semiclassical regime.

Abstract

We study the fluctuations of the diagonal matrix elements of the quantum cat map about their limit. We show that after suitable normalization, the fifth centered moment for the Hecke basis vanishes in the semiclassical limit, confirming in part a conjecture of Kurlberg and Rudnick. We also study sums of matrix elements lying in short windows. For observables with zero mean, the first moment of these sums is zero, and the variance was determined by the author with Kurlberg and Rudnick. We show that if the window is sufficiently small in terms of Planck's constant, the third moment vanishes if we normalize so that the variance is of order one.