Quenched Invariance Principle for a class of random conductance models\n with long-range jumps

2014/11/29 by Marek Biskup, Takashi Kumagai, Biskup, Marek +1
Mathematics · #Stochastic processes and statistical mechanics #Markov Chains and Monte Carlo Methods #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.1412.0175

Abstract

We study random walks on mathbb Z^d among random conductances\n C_xy colon x,y\∈ mathbb Z^d that permit jumps of arbitrary length.\nApart from joint ergodicity with respect to spatial shifts, we assume only that\nthe nearest-neighbor conductances are uniformly positive and that\n\∑_{x\∈ mathbb Z^d} C_0x|x|² is integrable. Our focus is on the\nQuenched Invariance Principle (QIP) which we establish in all d\≥3 by a\ncombination of corrector methods and heat-kernel technology. In particular, a\nQIP thus holds for random walks on long-range percolation graphs with exponents\nlarger than d+2 in all d\≥3, provided all nearest-neighbor edges are\npresent. We then show that, for long-range percolation with exponents between\nd+2 and 2d, the corrector fails to be sublinear everywhere. Similar\nexamples are constructed also for nearest-neighbor, ergodic conductances in\nd\≥4 under the conditions close to, albeit not exactly, complementary to\nthose of the recent work of S. Andres, M. Slowik and J.-D. Deuschel.\n

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