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We study notions of hyperuniformity for invariant locally square-integrable point processes in regular trees. We show that such point processes are never geometrically hyperuniform, and if the diffraction measure has support in the…
Self-similar Markov trees constitute a remarkable family of random compact real trees carrying a decoration function that is positive on the skeleton. As the terminology suggests, they are self-similar objects that further satisfy a Markov…
The spectral moments of ensembles of sparse random block matrices are analytically evaluated in the limit of large order. The structure of the sparse matrix corresponds to the Erd\"os-Renyi random graph. The blocks are i.i.d. random…
Two microring resonators, one with gain and one with loss, coupled to each other and to a bus waveguide, create an effective non-Hermitian potential for light propagating in the waveguide. Due to geometry, coupling for each microring…
Trees are partial orders in which every element has a linearly ordered set of predecessors. Here we initiate the exploration of the structural theory of trees with the study of different notions of \emph{branching in trees} and of…
In this work, we develop a spectral theory for hypergraph limits. We prove the convergence of the spectra of adjacency and Laplacian matrices for hypergraph sequences converging in the $1$-cut metric. On the other hand, we give examples of…
We calculate the mean and almost-sure leading order behaviour of the high frequency asymptotics of the eigenvalue counting function associated with the natural Dirichlet form on $\alpha$-stable trees, which lead in turn to short-time heat…
We derive exact equations for the spectral density of sparse networks with an arbitrary distribution of the number of single edges and triangles per node. These equations enable a systematic investigation of the effect of clustering on the…
Exceptional point and spectral singularity are two types of singularity that are unique to non-Hermitian systems. Here, we report the high-order spectral singularity as a high-order pole of the scattering matrix for a non-Hermitian…
Large spatial datasets often represent a number of spatial point processes generated by distinct entities or classes of events. When crossed with covariates, such as discrete time buckets, this can quickly result in a data set with millions…
Given a finite metric, one can construct its tight span, a geometric object representing the metric. The dimension of a tight span encodes, among other things, the size of the space of explanatory trees for that metric; for instance, if the…
We calculate the exact number of contours of size $n$ containing a fixed vertex in $d$-ary trees and provide sharp estimates for this number for more general trees. We also obtain a characterization of the locally finite trees with…
A $k$-tree is a spanning tree in which every vertex has degree at most $k$. In this paper, we provide a sufficient condition for the existence of a $k$-tree in a connected graph with fixed order in terms of the adjacency spectral radius and…
There is a rising interest in mapping trees using satellite or aerial imagery, but there is no standardized evaluation protocol for comparing and enhancing methods. In dense canopy areas, the high variability of tree sizes and their spatial…
We develop a unified spectral framework for finite ultrametric phylogenetic trees, grounding the analysis of phylogenetic structure in operator theory and stochastic dynamics in the finite setting. For a given finite ultrametric measure…
Every finite metric tree has generalized roundness strictly greater than one. On the other hand, some countable metric trees have generalized roundness precisely one. The purpose of this paper is to identify some large classes of countable…
The secular manifold $\Sigma_G$ and its singularities are intimately related to the spectra of metric graphs $(G,\ell)$. In this paper, we present a complete description of the singular locus for tree graphs, and confirm that it agrees with…
Scattering amplitudes for discrete states in 2D string theory are considered. Pole divergences of tree-level amplitudes are extracted and residues are interpreted as renormalized amplitudes for discrete states. An effective Lagrangian…
Let $T$ be a tree. Suppose $\lambda$ is an eigenvalue of the Laplacian matrix of $T$ with multiplicity $m_{T}(\lambda)$. It is known that $m_{T}(\lambda) \leq p(T)-1$, where $p(T)$ is the number of pendant vertices of $T$. In this paper, we…
We derive exact equations that determine the spectra of undirected and directed sparsely connected regular graphs containing loops of arbitrary length. The implications of our results to the structural and dynamical properties of networks…