Nearly Tight Bounds for Sandpile Transience on the Grid

We use techniques from the theory of electrical networks to give nearly tight bounds for the transience class of the Abelian sandpile model on the two-dimensional grid up to polylogarithmic factors. The Abelian sandpile model is a discrete process on graphs that is intimately related to the phenomenon of self-organized criticality. In this process, vertices receive grains of sand, and once the number of grains exceeds their degree, they topple by sending grains to their neighbors. The transience class of a model is the maximum number of grains that can be added to the system before it necessarily reaches its steady-state behavior or, equivalently, a recurrent state. Through a more refined and global analysis of electrical potentials and random walks, we give an $O(n^4\log^4{n})$ upper bound and an $\Omega(n^4)$ lower bound for the transience class of the $n \times n$ grid. Our methods naturally extend to $n^d$-sized $d$-dimensional grids to give $O(n^{3d - 2}\log^{d+2}{n})$ upper bounds and $\Omega(n^{3d -2})$ lower bounds.Comment: 36 pages, 4 figure

A novel Recurrence-Transience transition and Tracy-Widom growth in a cellular automaton with quenched noise

We study the growing patterns formed by a deterministic cellular automaton, the rotor-router model, in the presence of quenched noise. By the detailed study of two cases, we show that: (a) the boundary of the pattern displays KPZ fluctuations with a Tracy-Widom distribution, (b) as one increases the amount of randomness, the rotor-router path undergoes a transition from a recurrent to a transient walk. This transition is analysed here for the first time, and it is shown that it falls in the 3D Anisotropic Directed Percolation universality class.Comment: 6 pages + 8 pages SI, updated version with some correction

Processes on Unimodular Random Networks

We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasi-transitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amenability from the known context of unimodular quasi-transitive graphs to the more general context of unimodular random networks. We give properties of a trace associated to unimodular random networks with applications to stochastic comparison of continuous-time random walk.Comment: 66 pages; 3rd version corrects formula (4.4) -- the published version is incorrect --, as well as a minor error in the proof of Proposition 4.10; 4th version corrects proof of Proposition 7.1; 5th version corrects proof of Theorem 5.1; 6th version makes a few more minor correction

Analysis and Maintenance of Graph Laplacians via Random Walks

Graph Laplacians arise in many natural and artificial contexts. They are linear systems associated with undirected graphs. They are equivalent to electric flows which is a fundamental physical concept by itself and is closely related to other physical models, e.g., the Abelian sandpile model. Many real-world problems can be modeled and solved via Laplacian linear systems, including semi-supervised learning, graph clustering, and graph embedding. More recently, better theoretical understandings of Laplacians led to dramatic improvements across graph algorithms. The applications include dynamic connectivity problem, graph sketching, and most recently combinatorial optimization. For example, a sequence of papers improved the runtime for maximum flow and minimum cost flow in many different settings. In this thesis, we present works that the analyze, maintain and utilize Laplacian linear systems in both static and dynamic settings by representing them as random walks. This combinatorial representation leads to better bounds for Abelian sandpile model on grids, the first data structures for dynamic vertex sparsifiers and dynamic Laplacian solvers, and network flows on planar as well as general graphs.Ph.D

Sandpile Prediction on Structured Undirected Graphs

We present algorithms that compute the terminal configurations for sandpile instances in $O(n \log n)$ time on trees and $O(n)$ time on paths, where $n$ is the number of vertices. The Abelian Sandpile model is a well-known model used in exploring self-organized criticality. Despite a large amount of work on other aspects of sandpiles, there have been limited results in efficiently computing the terminal state, known as the sandpile prediction problem. Our algorithm improves the previous best runtime of $O(n \log^5 n)$ on trees [Ramachandran-Schild SODA '17] and $O(n \log n)$ on paths [Moore-Nilsson '99]. To do so, we move beyond the simulation of individual events by directly computing the number of firings for each vertex. The computation is accelerated using splittable binary search trees. We also generalize our algorithm to adapt at most three sink vertices, which is the first prediction algorithm faster than mere simulation on a sandpile model with sinks. We provide a general reduction that transforms the prediction problem on an arbitrary graph into problems on its subgraphs separated by any vertex set $P$. The reduction gives a time complexity of $O(\log^{|P|} n \cdot T)$ where $T$ denotes the total time for solving on each subgraph. In addition, we give algorithms in $O(n)$ time on cliques and $O(n \log^2 n)$ time on pseudotrees.Comment: 66 pages, submitted to SODA2

The roles of random boundary conditions in spin systems

Random boundary conditions are one of the simplest realizations of quenched disorder. They have been used as an illustration of various conceptual issues in the theory of disordered spin systems. Here we review some of these result