3/2 Firefighters are not enough

The firefighter problem is a monotone dynamic process in graphs that can be viewed as modeling the use of a limited supply of vaccinations to stop the spread of an epidemic. In more detail, a fire spreads through a graph, from burning vertices to their unprotected neighbors. In every round, a small amount of unburnt vertices can be protected by firefighters. How many firefighters per turn, on average, are needed to stop the fire from advancing? We prove tight lower and upper bounds on the amount of firefighters needed to control a fire in the Cartesian planar grid and in the strong planar grid, resolving two conjectures of Ng and Raff.Comment: 8 page

Forest-fire models and their critical limits

Forest-fire processes were first introduced in the physics literature as a toy model for self-organized criticality. The term self-organized criticality describes interacting particle systems which are governed by local interactions and are inherently driven towards a perpetual critical state. As in equilibrium statistical physics, the critical state is characterized by long-range correlations, power laws, fractal structures and self-similarity. We study several different forest-fire models, whose common features are the following: All models are continuous-time processes on the vertices of some graph. Every vertex can be "vacant" or "occupied by a tree". We start with some initial configuration. Then the process is governed by two competing random mechanisms: On the one hand, vertices become occupied according to rate 1 Poisson processes, independently of one another. On the other hand, occupied clusters are "set on fire" according to some predefined rule. In this case the entire cluster is instantaneously destroyed, i.e. all of its vertices become vacant. The self-organized critical behaviour of forest-fire models can only occur on infinite graphs such as planar lattices or infinite trees. However, in all relevant versions of forest-fire models, the destruction mechanism is a priori only well-defined for finite graphs. For this reason, one starts with a forest-fire model on finite subsets of an infinite graph and then takes the limit along increasing sequences of finite subsets to obtain a new forest-fire model on the infinite graph. In this thesis, we perform this kind of limit for two classes of forest-fire models and investigate the resulting limit processes

Determining Genus From Sandpile Torsor Algorithms

We provide a pair of ribbon graphs that have the same rotor routing and Bernardi sandpile torsors, but different topological genus. This resolves a question posed by M. Chan [Cha]. We also show that if we are given a graph, but not its ribbon structure, along with the rotor routing sandpile torsors, we are able to determine the ribbon graph's genus.Comment: Extended Abstract Accepted to FPSAC 2018. Revision of previous versio

Towards a four-loop form factor

The four-loop, two-point form factor contains the first non-planar correction to the lightlike cusp anomalous dimension. This anomalous dimension is a universal function which appears in many applications. Its planar part in N = 4 SYM is known, in principle, exactly from AdS/CFT and integrability while its non-planar part has been conjectured to vanish. The integrand of the form factor of the stress-tensor multiplet in N = 4 SYM including the non-planar part was obtained in previous work. We parametrise the difficulty of integrating this integrand. We have obtained a basis of master integrals for all integrals in the four-loop, two-point class in two ways. First, we computed an IBP reduction of the integrand of the N = 4 form factor using massive computer algebra (Reduze). Second, we computed a list of master integrals based on methods of the Mint package, suitably extended using Macaulay2 / Singular. The master integrals obtained in both ways are consistent with some minor exceptions. The second method indicates that the master integrals apply beyond N = 4 SYM, in particular to QCD. The numerical integration of several of the master integrals will be reported and remaining obstacles will be outlinedComment: 9 Pages, Radcor/Loopfest 2015 Proceeding

Efficient Dynamic Approximate Distance Oracles for Vertex-Labeled Planar Graphs

Let $G$ be a graph where each vertex is associated with a label. A Vertex-Labeled Approximate Distance Oracle is a data structure that, given a vertex $v$ and a label $\lambda$, returns a $(1+\varepsilon)$-approximation of the distance from $v$ to the closest vertex with label $\lambda$ in $G$. Such an oracle is dynamic if it also supports label changes. In this paper we present three different dynamic approximate vertex-labeled distance oracles for planar graphs, all with polylogarithmic query and update times, and nearly linear space requirements