Ihara Zeta Function Research Articles

Phases and Duality in the Fundamental Kazakov–Migdal Model on the Graph

Abstract We examine the fundamental Kazakov–Migdal (FKM) model on a generic graph, whose partition function is represented by the Ihara zeta function weighted by unitary matrices. The FKM model becomes unstable in the critical strip of the Ihara zeta function. We discover a duality between small and large couplings, associated with the functional equation of the Ihara zeta function for regular graphs. Although the duality is not precise for irregular graphs, we show that the effective action in the large coupling region can be represented by a summation of all possible Wilson loops on a graph similar to that in the small coupling region. We estimate the phase structure of the FKM model in both the small and large coupling regions by comparing it with the Gross–Witten–Wadia model. We further validate the theoretical analysis through detailed numerical simulations.

Spectral mapping theorem of an abstract non-unitary quantum walk

This paper continues the previous work (Quantum Inf. Process 11 (2019)) by two authors of the present paper about a spectral mapping property of chiral symmetric unitary operators. In physics, they treat non-unitary time-evolution operators to consider quantum walks in open systems. In this paper, we generalize the above result to include a chiral symmetric non-unitary operator whose coin operator only has two eigenvalues. As a result, the spectra of such non-unitary operators are included in the (possibly non-unit) circle and the real axis in the complex plane. We also give some examples of our abstract results, such as non-unitary quantum walks defined by Mochizuki et al. Furthermore, the main theorem applies not only to quantum walks but also to the Ihara zeta function and correlated random walks on regular graphs.

Quantum entanglement & purity testing: A graph zeta function perspective

We assign an arbitrary density matrix to a weighted graph and associate to it a graph zeta function that is both a generalization of the Ihara zeta function and a special case of the edge zeta function. We show that a recently developed bipartite pure state separability algorithm based on the symmetric group is equivalent to the condition that the coefficients in the exponential expansion of this zeta function are unity. Moreover, there is a one-to-one correspondence between the nonzero eigenvalues of a density matrix and the singularities of its zeta function. Several examples are given to illustrate these findings.

Walk/Zeta Correspondence

Our previous work presented explicit formulas for the generalized zeta function and the generalized Ihara zeta function corresponding to the Grover walk and the positive-support version of the Grover walk on the regular graph via the Konno–Sato theorem, respectively. This paper extends these walks to a class of walks including random walks, correlated random walks, quantum walks, and open quantum random walks on the torus by the Fourier analysis.

Graph zeta functions and Wilson loops in a Kazakov–Migdal model

Abstract In this paper, we consider an extended Kazakov–Migdal model defined on an arbitrary graph. The partition function of the model, which is expressed as the summation of all Wilson loops on the graph, turns out to be represented by the Bartholdi zeta function weighted by unitary matrices on the edges of the graph. The partition function on the cycle graph at finite N is expressed by the generating function of the generalized Catalan numbers. The partition function on an arbitrary graph can be exactly evaluated at large N, which is expressed as an infinite product of a kind of deformed Ihara zeta function. The non-zero-area Wilson loops do not contribute to the leading part of the 1/N expansion of the free energy but to the next leading. The semi-circle distribution of the eigenvalues of the scalar fields is still an exact solution of the model at large N on an arbitrary regular graph, but it reflects only zero-area Wilson loops.

Kazakov-Migdal model on the graph and Ihara zeta function

We propose the Kazakov-Migdal model on graphs and show that, when the parameters of this model are appropriately tuned, the partition function is represented by the unitary matrix integral of an extended Ihara zeta function, which has a series expansion by all non-collapsing Wilson loops with their lengths as weights. The partition function of the model is expressed in two different ways according to the order of integration. A specific unitary matrix integral can be performed at any finite N thanks to this duality. We exactly evaluate the partition function of the parameter-tuned Kazakov-Migdal model on an arbitrary graph in the large N limit and show that it is expressed by the infinite product of the Ihara zeta functions of the graph.

A generalized Ihara zeta function formula for simple graphs with bounded degree

We establish a generalized Ihara zeta function formula for simple graphs with bounded degree. This is a generalization of the formula obtained by G. Chinta, J. Jorgenson and A. Karlsson from vertex-transitive graphs.

On Sidorenko's conjecture for determinants and Gaussian Markov random fields

AbstractWe study a class of determinant inequalities that are closely related to Sidorenko's famous conjecture (also conjectured by Erdős and Simonovits in a different form). Our main result can also be interpreted as an entropy inequality for Gaussian Markov random fields (GMRF). We call a GMRF on a finite graph homogeneous if the marginal distributions on the edges are all identical. We show that if is bipartite, then the differential entropy of any homogeneous GMRF on is at least times the edge entropy plus times the point entropy. We also show that in the case of non‐negative correlation on edges, the result holds for an arbitrary graph . The connection between Sidorenko's conjecture and GMRF's is established via a large deviation principle on high dimensional spheres combined with graph limit theory. It is also observed that the system we study exhibits a phase transition on large girth regular graphs. Connection with Ihara zeta function and the number of spanning trees is also discussed.

Asymptotic Absence of Poles of Ihara Zeta Function of Large Erdős–Rényi Random Graphs

Asymptotic Absence of Poles of Ihara Zeta Function of Large Erdős–Rényi Random Graphs

The Ihara-Zeta Function and the Spectrum of the Join of Two Semi-Regular Bipartite Graphs

In this paper, using matrix techniques, we compute the Ihara-zeta function and the number of spanning trees of the join of two semi-regular bipartite graphs. Furthermore, we show that the spectrum and the Ihara-zeta function of the join of two semi-regular bipartite graphs can determine each other.

Kirchhoff’s theorem for Prym varieties

Abstract We prove an analogue of Kirchhoff’s matrix tree theorem for computing the volume of the tropical Prym variety for double covers of metric graphs. We interpret the formula in terms of a semi-canonical decomposition of the tropical Prym variety, via a careful study of the tropical Abel–Prym map. In particular, we show that the map is harmonic, determine its degree at every cell of the decomposition and prove that its global degree is $2^{g-1}$ . Along the way, we use the Ihara zeta function to provide a new proof of the analogous result for finite graphs. As a counterpart, the appendix by Sebastian Casalaina-Martin shows that the degree of the algebraic Abel–Prym map is $2^{g-1}$ as well.

The Ihara zeta function of the complement of a semiregular bipartite graph

We study the Ihara zeta function of the complement of a semiregular bipartite graph. A factorization formula for the Ihara zeta function is derived via which the number of spanning trees is computed. For a class of complements of semiregular bipartite graphs, it is shown that they have the same Ihara zeta function if and only if they are cospectral.

Grover/Zeta Correspondence based on the Konno–Sato theorem

Recently the Ihara zeta function for the finite graph was extended to infinite one by Clair and Chinta et al. In this paper, we obtain the same expressions by a different approach from their analytical method. Our new approach is to take a suitable limit of a sequence of finite graphs via the Konno-Sato theorem. This theorem is related to explicit formulas of characteristic polynomials for the evolution matrix of the Grover walk. The walk is one of the most well-investigated quantum walks which are quantum counterpart of classical random walks. We call the relation between the Grover walk and the zeta function based on the Konno-Sato theorem "Grover/Zeta Correspondence" here.

Zeta functions of edge-free quotients of graphs

We define the Ihara zeta function ζ(u,X) and Artin–Ihara L-function of the quotient graph of groups X=Y//G, where G is a group acting on a finite graph Y with trivial edge stabilizers. We determine the relationship between the primes of Y and X and show that the projection map Y→X can be naturally viewed as an unramified Galois covering of graphs of groups. We show that the L-function of X evaluated at the regular representation is equal to ζ(u,Y), and that ζ(u,X) divides ζ(u,Y). We derive two-term and three-term determinant formulas for the zeta and L-functions, and compute several examples of L-functions of edge-free quotients of the tetrahedron graph K4.

A note on generalized semitotal point graphs

Let Rsk(G) be the graph obtained from a graph G by gluing k distinct path graphs Ps+2 to both vertices incident to each edge of G. In this paper, we derive a factorization formula for the generalized characteristic polynomial of the graph Rsk(G), where G is regular. As particular cases, we obtain formulas for the characteristic polynomials of the adjacency, Laplacian, normalized Laplacian, and signless Laplacian matrices of Rsk(G), as well as for the Ihara zeta function of Rsk(G), where G is regular. We also derive formulas for the Kirchhoff, additive degree-Kirchhoff, and multiplicative degree-Kirchhoff indices of Rsk(G), where G is regular. In addition, we determine the weighted Kirchhoff index of Rsk(G), for an arbitrary graph G. This work generalizes recent results on semitotal point graphs.

Ihara Zeta function, coefficients of Maclaurin series and Ramanujan graphs

Let [Formula: see text] denote a connected [Formula: see text]-regular undirected graph of finite order [Formula: see text]. The graph [Formula: see text] is called Ramanujan whenever [Formula: see text] for all nontrivial eigenvalues [Formula: see text] of [Formula: see text]. We consider the variant [Formula: see text] of the Ihara Zeta function [Formula: see text] of [Formula: see text] defined by [Formula: see text] The function [Formula: see text] satisfies the functional equation [Formula: see text]. Let [Formula: see text] denote the number sequence given by [Formula: see text] In this paper, we establish the equivalence of the following statements: (i) [Formula: see text] is Ramanujan; (ii) [Formula: see text] for all [Formula: see text]; (iii) [Formula: see text] for infinitely many even [Formula: see text]. Furthermore, we derive the Hasse–Weil bound for the Ramanujan graphs.

Ihara zeta functions of coronae

We derive a factorization formula for the Ihara zeta function of the corona of two graphs. When the two graphs are regular, this factorization formula involves the graphs' adjacency spectra. As an application, we show that if each composing graph of a corona of regular graphs is replaced with a cospectral mate, then the Ihara zeta function of the corona does not change. We also give a formula for the weighted Kirchhoff index of the corona of two regular graphs.

Ihara zeta function of finite graphs with circuit rank two

In this paper, we give an explicit formula as a rational function for the Ihara zeta function of every finite connected graph without degree one vertices whose circuit rank is two.

Ihara Zeta Function and Spectrum of the Cone Over a Semiregular Bipartite Graph

In this paper, a formula for the Ihara zeta function of the cone over a semiregular bipartite graph is derived. Using this formula, we show that two cones over semiregular bipartite graphs are cospectral if and only if they have the same Ihara zeta function. Moreover, the convergence of the zeta function of this family of graphs is considered.

Building lattices and zeta functions

AbstractWe give a Lefschetz formula for tree lattices and apply it to geometric zeta functions. We further generalize Bass’s approach to Ihara zeta functions to the higher-dimensional case of a building.