Noise sensitivity and exceptional times of transience for a simple symmetric random walk in one dimension

Abstract

We define a dynamical simple symmetric random walk in one dimension, and show that there almost surely exist exceptional times at which the walk tends to infinity. This is in contrast to the usual dynamical simple symmetric random walk in one dimension, for which such exceptional times are known not to exist. In fact we show that the set of exceptional times has Hausdorff dimension 1/2 almost surely, and give bounds on the rate at which the walk diverges at such times. We also show noise sensitivity of the event that our random walk is positive after n steps. In fact this event is maximally noise sensitive, in the sense that it is quantitatively noise sensitive for any sequence $$\varepsilon _n$$ ε n such that $$n\varepsilon _n\rightarrow \infty $$ n ε n → ∞ . This is again in contrast to the usual random walk, for which the corresponding event is known to be noise stable.

The probability of unusually large components for critical percolation on random $d$-regular graphs

“…The first statement follows from a result of Ritter [38]; see [36,Lemma 8] for details. The second statement follows from results of [17], specifically Theorem 3.10 (a general theorem on convergence of conditioned Markov processes) together with the results of Section 4.2 (where it is shown that finite variance, mean-zero random walks satisfy the conditions of the earlier theorem).…”

Limit theorems and ergodicity for general bootstrap random walks

“…In mathematical finance, it was used to discuss the continuity of utility maximization under weak convergence (see [1]). Within the context of noise sensitivity, [15] compares the effects on the sequences (X n ) n≥0 and (Y n ) n≥0 of "Poisson switches" in the sequence (ξ n ) n≥1 . Owing to the fact that every switch in the sequence (ξ n ) n≥1 results in multiple concurrent switches in the sequence (η n ) n≥1 , it is shown in [15] that the noise sensitivity of (Y n ) n≥0 is greater than that of (X n ) n≥0 .…”