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Register a new userThe abelian sandpile model on a random binary tree
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We study the abelian sandpile model on a random binary tree. Using a transfer matrix approach introduced by Dhar & Majumdar, we prove exponential decay of correlations, and in a small supercritical region (i.e., where the branching process survives with positive probability) exponential decay of avalanche sizes. This shows a phase transition phenomenon between exponential decay and power law decay of avalanche sizes. Our main technical tools are: (1) A recursion for the ratio between the numbers of weakly and strongly allowed configurations which is proved to have a well-defined stochastic solution; (2) quenched and annealed estimates of the eigenvalues of a product of $n$ random transfer matrices.
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Consider an infinite tree with random degrees, i.i.d. over the sites, with a prescribed probability distribution with generating function G(s). We consider the following variation of Renyis parking problem, alternatively called blocking RSA: at every vertex of the tree a particle (or car) arrives with rate one. The particle sticks to the vertex whenever the vertex and all of its nearest neighbors are not occupied yet. We provide an explicit expression for the so-called parking constant in terms of the generating function.
We introduce a natural stochastic extension, called SSP, of the abelian sandpile model(ASM), which shares many mathematical properties with ASM, yet radically differs in its physical behavior, for example in terms of the shape of the steady state and of the avalanche size distribution. We establish a basic theory of SSP analogous to that of ASM, and present a brief numerical study of its behavior. Our original motivation for studying SSP stems from its connection to the LLL algorithm established in another work by the authors [5]. The importance of understanding how LLL works cannot be stressed more, especially from the point of view of lattice-based cryptography. We believe SSP serves as a tractable toy model of LLL that would help further our understanding of it.
In a recent paper [15], Giardin{`a}, Giberti, Hofstad, Prioriello have proved a law of large number and a central limit theorem with respect to the annealed measure for the magnetization of the Ising model on some random graphs including the random 2-regular graph. We present a new proof of their results, which applies to all random regular graphs. In addition, we prove the existence of annealed pressure in the case of configuration model random graphs.
For $Delta ge 5$ and $q$ large as a function of $Delta$, we give a detailed picture of the phase transition of the random cluster model on random $Delta$-regular graphs. In particular, we determine the limiting distribution of the weights of the ordered and disordered phases at criticality and prove exponential decay of correlations and central limit theorems away from criticality. Our techniques are based on using polymer models and the cluster expansion to control deviations from the ordered and disordered ground states. These techniques also yield efficient approximate counting and sampling algorithms for the Potts and random cluster models on random $Delta$-regular graphs at all temperatures when $q$ is large. This includes the critical temperature at which it is known the Glauber and Swendsen-Wang dynamics for the Potts model mix slowly. We further prove new slow-mixing results for Markov chains, most notably that the Swendsen-Wang dynamics mix exponentially slowly throughout an open interval containing the critical temperature. This was previously only known at the critical temperature. Many of our results apply more generally to $Delta$-regular graphs satisfying a small-set expansion condition.
We consider ferromagnetic Ising models on graphs that converge locally to trees. Examples include random regular graphs with bounded degree and uniformly random graphs with bounded average degree. We prove that the cavity prediction for the limiting free energy per spin is correct for any positive temperature and external field. Further, local marginals can be approximated by iterating a set of mean field (cavity) equations. Both results are achieved by proving the local convergence of the Boltzmann distribution on the original graph to the Boltzmann distribution on the appropriate infinite random tree.
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