A shape theorem and a variational formula for the quenched Lyapunov exponent of random walk in a random potential

Janjigian, Christopher; Sergazy, Nurbavliyev,; Rassoul-Agha, Firas

doi:10.48550/arxiv.2006.10871

Cited by 2 publications

(3 citation statements)

References 32 publications

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“…potentials and the simple random walk on Z d in the present and aforementioned articles, let us finally refer results for models with various changes of our setting. In [8,17], the underlying space is Z d , but the potential is stationary and ergodic and each step of the random walk is in an arbitrary finite set. Under such a more general setting, [17] studied the quenched large deviation principle and [8] constructed the quenched Lyapunov exponent.…”

Section: Introductionmentioning

confidence: 99%

“…In [8,17], the underlying space is Z d , but the potential is stationary and ergodic and each step of the random walk is in an arbitrary finite set. Under such a more general setting, [17] studied the quenched large deviation principle and [8] constructed the quenched Lyapunov exponent. On the other hand, [18, Part II] treats a Brownian motion evolving in a Poissonian potential, which is a continuum version of our model.…”

Section: Introductionmentioning

confidence: 99%

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Strict comparison for the Lyapunov exponents of the simple random walk in random potentials

Kubota¹

2020

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We consider the simple random walk in i.i.d. nonnegative potentials on the d-dimensional cubic lattice Z d (d ≥ 1). In this model, the so-called Lyapunov exponent describes the cost of traveling for the simple random walk in the potential. The Lyapunov exponent depends on the distribution function of the potential, and the aim of this article is to prove that the Lyapunov exponent is strictly monotone in the distribution function of the potential with the order according to strict dominance. In particular, for the one-dimensional annealed situation, we observe that the Lyapunov exponents can coincide even under the strict dominance. Furthermore, the comparison for the Lyapunov exponent also provides that for the rate function of this model.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Strict comparison for the Lyapunov exponents of the simple random walk in random potentials

Kubota¹

2020

Preprint

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“…The "cocycle" branch of [52,51,54,28] has also been developed for FPP [34,43] and bears connections to the formula of Krishnan. Meanwhile, the "entropy" branch led to an LPP formula [28,Thm.…”

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confidence: 99%

Empirical Measures, Geodesic Lengths, and a Variational Formula in First-Passage Percolation

Bates

2020

Preprint

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This article resolves, in a dense set of cases, several open problems concerning geodesics in i.i.d. first-passage percolation on Z d . Our primary interest is in the empirical measures of edge-weights observed along geodesics from 0 to nξ, where ξ is a fixed unit vector. For various dense families of edge-weight distributions, we prove that these measures converge weakly to a deterministic limit as n → ∞, answering a question of Hoffman. These families include arbitrarily small L ∞ -perturbations of any given distribution, almost every finitely supported distribution, uncountable collections of continuous distributions, and certain discrete distributions whose atoms have any prescribed sequence of probabilities. Moreover, the constructions are explicit enough to guarantee examples possessing certain features, for instance: both continuous and discrete distributions whose support is all of [0, ∞), and distributions given by a density function that is k-times differentiable. All results also hold for ξ-directed infinite geodesics. In comparison, we show that if Z d is replaced by the infinite d-ary tree, then any weight distribution admits a unique limiting empirical measure along geodesics. In both the lattice and tree cases, our methodology is driven by a new variational formula for the time constant, which requires no assumptions on the edge-weight distribution. Incidentally, this variational approach also allows us to obtain new convergence results for geodesic lengths, which have been unimproved in the subcritical case since the seminal 1965 manuscript of Hammersley and Welsh. Contents 0. Outline of paper 1. Introduction: definitions and main questions 2. Variational formula for the time constant 3. Applications of variational formula: main results and their proofs 4. First-passage percolation on d-ary tree 5. Negative weights and passage times along geodesics 6. Construction of the constraint set 7. Proof of variational formula 8. Proof of empirical measure convergence in tree case 9. Acknowledgments References List of symbols

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A shape theorem and a variational formula for the quenched Lyapunov exponent of random walk in a random potential

Cited by 2 publications

References 32 publications

Strict comparison for the Lyapunov exponents of the simple random walk in random potentials

Strict comparison for the Lyapunov exponents of the simple random walk in random potentials

Empirical Measures, Geodesic Lengths, and a Variational Formula in First-Passage Percolation

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