Nearly Tight Bounds for Sandpile Transience on the Grid

Durfee, David; Fahrbach, Matthew; Gao, Yu; Xiao, Tao

doi:10.1137/1.9781611975031.40

Cited by 4 publications

(4 citation statements)

References 16 publications

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“…The dynamics of a single lattice path are equivalent to a those of a correlated random walk, so in Section 5 we develop a new tail inequality for correlated random walks that accurately bounds the probability of large deviations from the starting position. We note that decomposing the dynamics of lattice models into one-dimensional random walks has recently been shown to achieve nearly-tight bounds for escape probabilities in di erent settings [DFGX18].…”

Section: Techniquesmentioning

confidence: 98%

Slow Mixing of Glauber Dynamics for the Six-Vertex Model in the Ordered Phases

Fahrbach,

Randall

2019

Preprint

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We analyze the mixing time of Glauber dynamics for the six-vertex model in the ferroelectric and antiferroelectric phases. Speci cally, we show that for all Boltzmann weights in the ferroelectric phase, there exist boundary conditions such that local Markov chains require exponential time to converge to equilibrium. This is the rst rigorous result about the mixing time of Glauber dynamics for the six-vertex model in the ferroelectric phase. Furthermore, our analysis demonstrates a fundamental connection between correlated random walks and the dynamics of intersecting lattice path models. We also analyze the Glauber dynamics for the six-vertex model with free boundary conditions in the antiferroelectric phase and signi cantly extend the region for which local Markov chains are known to be slow mixing. This result relies on a Peierls argument and novel properties of weighted non-backtracking walks.

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Section: Techniquesmentioning

confidence: 98%

Slow Mixing of Glauber Dynamics for the Six-Vertex Model in the Ordered Phases

Fahrbach,

Randall

2019

Preprint

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“…Specifically, that the unit electrical flow from s to t is the expected trajectory of the random walk from s to t, with cancellations, where from vertex u we go to v ∼ u with probability r −1 uv w∼u r −1 uw where the reciprocal of resistances, conductance, plays a role analogous to the weight of edges. Many of our intuitions and notations have overlaps with the electrical flow based analyses of sandpile processes [DFGX18]. For a more systematic exposition, we refer the reader to the excellent monograph by Doyle and Snell [DS84].…”

Section: Electrical Flow and Random Walksmentioning

confidence: 99%

“…A variant of this notation was also central to [DFGX18]. We choose the use the starting vector as a parameter because we will extend it to a vector on all vertices below in Definition 2.6.…”

Section: Schur Complementsmentioning

confidence: 99%

Fully Dynamic Electrical Flows: Sparse Maxflow Faster Than Goldberg-Rao

Gao¹,

Liu²,

Peng³

2021

Preprint

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We give an algorithm for computing exact maximum flows on graphs with m edges and integer capacities in the rangeFor sparse graphs with polynomially bounded integer capacities, this is the first improvement over the O(m 1.5 log U ) time bound from [Goldberg-Rao JACM '98].Our algorithm revolves around dynamically maintaining the augmenting electrical flows at the core of the interior point method based algorithm from [Mądry JACM '16]. This entails designing data structures that, in limited settings, return edges with large electric energy in a graph undergoing resistance updates.

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“…The effective resistance of an edge is a fundamental quantity. It and its extensions have a variety of connections in the analysis of networks [SM07,Sar10], combinatorics [Lov93,DFGX18] and the design of better graph algorithms [CKM + 11, MST15,Sch17].…”

Section: Estimating Effective-resistancesmentioning

confidence: 99%

Graph Sparsification, Spectral Sketches, and Faster Resistance Computation, via Short Cycle Decompositions

Chu

Gao

Peng

et al. 2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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We develop a framework for graph sparsification and sketching, based on a new tool, short cycle decomposition -a decomposition of an unweighted graph into an edge-disjoint collection of short cycles, plus a small number of extra edges. A simple observation gives that every graph G on n vertices with m edges can be decomposed in O(mn) time into cycles of length at most 2 log n, and at most 2n extra edges. We give an m 1+o(1) time algorithm for constructing a short cycle decomposition, with cycles of length n o(1) , and n 1+o(1) extra edges. Both the existential and algorithmic variants of this decomposition enable us to make progress on several open problems in randomized graph algorithms.1. We present an algorithm that runs in time m 1+o(1) ε −1.5 and returns (1±ε)-approximations to effective resistances of all edges, improving over the previous best of O (min{mε −2 , n 2 ε −1 }). This routine in turn gives an algorithm to approximate the determinant of a graph Laplacian up to a factor of (1 ± ε) in m 1+o(1) + n 15 /8+o(1) ε − 7 /4 time.2. We show existence and efficient algorithms for constructing graphical spectral sketchesa distribution over sparse graphs H such that for a fixed vector x , we havewhere L is the graph Laplacian and L + is its pseudoinverse. This implies the existence of resistance-sparsifiers with about nε −1 edges that preserve the effective resistances between every pair of vertices up to (1 ± ε). By combining short cycle decompositions with known tools in graph sparsification, weshow the existence of nearly-linear sized degree-preserving spectral sparsifiers, as well as significantly sparser approximations of directed graphs. The latter is critical to recent breakthroughs on faster algorithms for solving linear systems in directed Laplacians.The running time and output qualities of our spectral sketch and degree-preserving (directed) sparsification algorithms are limited by the efficiency of our routines for constructing short cycle decompositions. Improved algorithms for short cycle decompositions will lead to improvements for each of these algorithms.2. What if we only want to preserve the effective resistance 2 between all pairs of vertices? Dinitz, Krauthgamer, and Wagner [DKW15] define such a graph H as a resistance sparsifier of G, and show their existence for regular expanders with degree Ω(n). They conjecture that every graph admits an ε-resistance sparsifier with O (nε −1 ) edges.3. An ε-spectral sparsifier preserves weighted vertex degrees up to (1±ε). Do there exist spectral sparsifiers that exactly preserve weighted degrees? Dinitz et al. [DKW15] also explicitly pose a related question -does every dense regular expander contain a sparse regular expander? 4. What about sparsification for directed graphs? The above sparsification notions, and algorithms are difficult to generalize to directed graphs. Cohen et al. [CKP + 16] developed a notion of sparsification for Eulerian directed graphs (directed graphs with all vertices having in-degree equal to out-degree), and gave the first a...

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Nearly Tight Bounds for Sandpile Transience on the Grid

Cited by 4 publications

References 16 publications

Slow Mixing of Glauber Dynamics for the Six-Vertex Model in the Ordered Phases

Slow Mixing of Glauber Dynamics for the Six-Vertex Model in the Ordered Phases

Fully Dynamic Electrical Flows: Sparse Maxflow Faster Than Goldberg-Rao

Graph Sparsification, Spectral Sketches, and Faster Resistance Computation, via Short Cycle Decompositions

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