arXiv subjects

Tori Hudgins

Publications and source records attributed to Tori Hudgins.

4 recordsLinked to original sources

arXiv2025

Periodic orbits on 2-regular circulant digraphs

Periodic orbits (equivalence classes of closed paths up to cyclic shifts) play an important role in applications of graph theory. For example, they appear in the definition of the Ihara zeta function and exact trace formulae for the spectra of quantum graphs. Circulant graphs are Cayley graphs of $\mathbb{Z}_n$. Here we consider directed Cayley graphs with two generators (2-regular Cayley digraphs). We determine the number of primitive periodic orbits of a given length (total number of directed edges) in terms of the number of times edges corresponding to each generator appear in the periodic orbit (the step count). Primitive periodic orbits are those periodic orbits that cannot be written as a repetition of a shorter orbit. We describe the lattice structure of lengths and step counts for which periodic orbits exist and characterize the repetition number of a periodic orbit by its winding number (the sum of the step sequence divided by the number of vertices) and the repetition number of its step sequence. To obtain these results, we also evaluate the number of Lyndon words on an alphabet of two letters with a given length and letter count.

math.CO↗

arXiv2022

Complete Dynamical Evaluation of the Characteristic Polynomial of Binary Quantum Graphs

We evaluate the variance of coefficients of the characteristic polynomial for binary quantum graphs using a dynamical approach. This is the first example where a spectral statistic can be evaluated in terms of periodic orbits for a system with chaotic classical dynamics without taking the semiclassical limit, which here is the limit of large graphs. The variance depends on the sizes of particular sets of primitive pseudo orbits (sets of distinct primitive periodic orbits): the set of primitive pseudo orbits without self-intersections and the sets of primitive pseudo orbits with a fixed number of self-intersections, all of which consist of two arcs of the pseudo orbit crossing at a single vertex. To show other pseudo orbits do not contribute we give two arguments. The first is based on a reduction of the variance formula from a sum over pairs of primitive pseudo orbits to a sum over pseudo orbits where no bonds are repeated. The second employs a parity argument for the Lyndon decomposition of words. For families of binary graphs, in the semiclassical limit, we show the pseudo orbit formula approaches a universal constant independent of the coefficient of the polynomial. This is obtained by counting the total number of primitive pseudo orbits of a given length.

math-ph↗

arXiv2022

Periodic orbit evaluation of a spectral statistic of quantum graphs without the semiclassical limit

Energy level statistics of quantized chaotic systems have been evaluated in the semiclassical limit via their periodic orbits using the Gutzwiller and related trace formulae. Here we evaluate a spectral statistic of chaotic 4-regular quantum graphs from their periodic orbits without the semiclassical limit. The variance of the n-th coefficient of the characteristic polynomial is determined by the sizes of the sets of distinct primitive periodic orbits with n bonds which have no self-intersections, and the sizes of the sets with a given number of self-intersections which all consist of two sections of the pseudo orbit crossing at a single vertex. Using this result we observe the mechanism that connects semiclassical results to the total number of orbits regardless of their structure.

quant-ph↗

arXiv2021

A combinatorial approach to counting primitive periodic and primitive pseudo orbits on circulant graphs

For families of 4-regular directed circulant graphs with $n$ vertices, we count the number of primitive periodic orbits of length up to at least $n$. The relevant counting techniques are then extended to count the number of primitive pseudo orbits (sets of distinct primitive periodic orbits) of length up to at least $n$ that lack self-intersections, or that self-intersect only at individual vertices repeated exactly twice (2-encounters of length zero), for two particular families of 4-regular directed circulant graphs. We then regard these two families of graphs as families of quantum graphs and use the counting results to compute the variance of the coefficients of the quantum graph's characteristic polynomial.

math.CO↗