相关论文: Spectral statistics for scaling quantum graphs
We introduce the notion of Benjamini-Schramm convergence for quantum graphs. This notion of convergence, intended to play the role of the already existing notion for discrete graphs, means that the restriction of the quantum graph to a…
Determining and analyzing the spectra of graphs is an important and exciting research topic in theoretical computer science. The eigenvalues of the normalized Laplacian of a graph provide information on its structural properties and also on…
In this paper, a new measurement to compare two large-scale graphs based on the theory of quantum probability is proposed. An explicit form for the spectral distribution of the corresponding adjacency matrix of a graph is established. Our…
In this Thesis, we study the behavior of spectral curves of quantum graphs under certain families of vertex conditions, called the $\delta_s$ family, which we define in this work. We focus on studying two main quantities related to the…
Quantum graphs have recently been introduced as model systems to study the spectral statistics of linear wave problems with chaotic classical limits. It is proposed here to generalise this approach by considering arbitrary, directed graphs…
It is well known that, in general, part of the spectrum of a graph can be obtained from the adjacency matrix of its quotient graph given by a regular partition. In this paper, we propose a method to obtain all the spectrum, and also the…
We review quantum chaos on graphs. We construct a unitary operator which represents the quantum evolution on the graph and study its spectral and wavefunction statistics. This operator is the analogue of the classical evolution operator on…
We reinterpret the spectral dimension of spacetimes as the scaling of an effective self-energy transition amplitude in quantum field theory (QFT), when the system is probed at a given resolution. This picture has four main advantages: (a)…
We study the transmission of a quantum particle along a straight input--output line to which a graph $\Gamma$ is attached at a point. In the point of contact we impose a singularity represented by a certain properly chosen scale-invariant…
We investigate the spectral properties of chaotic quantum graphs. We demonstrate that the `energy'--average over the spectrum of individual graphs can be traded for the functional average over a supersymmetric non--linear $\sigma$--model…
We propose to analyse the statistical properties of a sequence of vectors using the spectrum of the associated Gram matrix. Such sequences arise e.g. by the repeated action of a deterministic kicked quantum dynamics on an initial condition…
A detailed discussion of semiclassical trace formulae is presented and it is demonstrated how a regularized trace formula can be derived while dealing only with finite and convergent expressions. Furthermore, several applications of trace…
We derive a formula for the level spacing probability distribution in quantum graphs. We apply it to simple examples and we discuss its relation with previous work and its possible application in more general cases. Moreover, we derive an…
A method for estimating the spectral gap along with higher eigenvalues of nonequilateral quantum graphs has been introduced by Amini and Cohen-Steiner recently: it is based on a new transference principle between discrete and continuous…
The spectral gap of local random quantum circuits is a fundamental property that determines how close the moments of the circuit's unitaries match those of a Haar random distribution. When studying spectral gaps, it is common to bound these…
We use random matrix theory to study the spectrum of random geometric graphs, a fundamental model of spatial networks. Considering ensembles of random geometric graphs we look at short range correlations in the level spacings of the…
Given an unbalanced open quantum graph, we derive a formula relating sums over its scattering resonances with integrals outside a strip. We deduce lower bounds on the number of resonances (in bounded regions of the complex plane),that are…
We prove that the spectrum of an individual chaotic quantum graph shows universal spectral correlations, as predicted by random--matrix theory. The stability of these correlations with regard to non--universal corrections is analyzed in…
Boundary conditions in quantum graph vertices are generally given in terms of a unitary matrix $U$. Observing that if $U$ has at most two eigenvalues, then the scattering matrix $\mathcal{S}(k)$ of the vertex is a linear combination of the…
We present quantum graphs with remarkably regular spectral characteristics. We call them {\it regular quantum graphs}. Although regular quantum graphs are strongly chaotic in the classical limit, their quantum spectra are explicitly…