Related papers: Chip-Firing and Rotor-Routing on Directed Graphs
In an ongoing project to classify all hereditary abelian categories, we provide a classification of Ext-finite directed hereditary abelian categories satisfying Serre duality up to derived equivalence. In order to prove the classification,…
We give a self-contained introduction to the theory of directed graphs, leading up to the relationship between the Perron-Frobenius eigenvectors of a graph and its autocatalytic sets. Then we discuss a particular dynamical system on a fixed…
The behavior of orbits in charged-particle beam transport systems, including both linear and circular accelerators as well as final focus sections and spectrometers, can depend sensitively on nonlinear fringe-field and high-order-multipole…
Motivated by the notion of chip-firing on the dual graph of a planar graph, we consider `integral flow chip-firing' on an arbitrary graph $G$. The chip-firing rule is governed by ${\mathcal L}^*(G)$, the dual Laplacian of $G$ determined by…
In this paper we derive the scaling fields in $c=-2$ conformal field theory associated with weakly allowed clusters in abelian sandpile model and show a direct relation between the two models.
We study orbit closures and stationary measures for groups of automorphisms of $p$-adic affine surfaces.
Let $G$ be an infinite graph such that each tree in the wired uniform spanning forest on $G$ has one end almost surely. On such graphs $G$, we give a family of continuous, measure preserving, almost one-to-one mappings from the wired…
In this paper, we consider the Ricci curvature of a directed graph, based on Lin-Lu-Yau's definition. We give some properties of the Ricci curvature, including conditions for a directed regular graph to be Ricci-flat. Moreover, we calculate…
We establish a connection between finite fields and finite dynamical systems. We show how this connection can be used to shed light on some problems in finite dynamical systems and in particular, in linear systems.
This work proposes an extension to the leader-first follower (LFF) class of graphs used to solve the bearing-only formation control problem over directed graphs. The first contribution provides an equilibrium, stability, and convergence…
In this paper we study the dynamics of rational maps induced by endomorphisms of ordinary elliptic curves defined over finite fields.
In the abelian sandpile model, recurrent chip configurations are of interest as they are a natural choice of coset representatives under the quotient of the reduced Laplacian. We investigate graphs whose recurrent identities with respect to…
We investigate a set of directed sandpile models on the Apollonian network, which are inspired on the work by Dhar and Ramaswamy (PRL \textbf{63}, 1659 (1989)) for Euclidian lattices. They are characterized by a single parameter $q$, that…
We consider the abelian sandpile model and the uniform spanning unicycle on random planar maps. We show that the sandpile density converges to 5/2 as the maps get large. For the spanning unicycle, we show that the length and area of the…
We numerically study the directed version of the fixed energy sandpile. On a closed square lattice, the dynamical evolution of a fixed density of sand grains is studied. The activity of the system shows a continuous phase transition around…
This paper considers the segregation of a granular mixture in a rotating drum. Extending a recent kinematic model for grain transport on sandpile surfaces to the case of rotating drums, an analysis is presented for radial segregation in the…
We consider fibrations by affine lines on smooth affine surfaces obtained as complements of smooth rational curves $B$ in smooth projective surfaces $X$ defined over an algebraically closed field of characteristic zero. We observe that…
We examine domino tilings of rectangular boards, which are in natural bijection with perfect matchings of grid graphs. This leads to the study of their associated perfect matching polytopes, and we present some of their properties, in…
We study the morphological evolution of surfaces during ion sputtering and we compare their dynamical roughening with aeolian ripple formation in sandy deserts. We show that, although the two phenomena are physically different, they must…
We investigate numerically the scattering of waves on discrete graphs. An efficient algorithm is developed to compute the reflection and transmission (spectral) coefficients. We then explore various configurations of input and output leads,…