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Quantitative noise sensitivity and exceptional times for percolation. (English) Zbl 1213.60160
In percolation theory, typical events of interest depend on a huge (or infinite) number of independent Bernoulli random variables. How does the outcome of such an event change if we allow a growing number of these random variables to change value each with small probability? The notion of noise sensitivity formalizes this question and allows for quantitative answers.
The authors derive noise sensitivity estimates for the event of an open left to right crossing in an (approximate) square box both for critical bond percolation on the two-dimensional integer lattice and for critical site percolation on the triangular lattice. A key step for the proofs is a bound on the Fourier coefficients of a function \(f(x)\) depending on the number of bits of \(x\) that a randomized algorithm needs in order to determine the value of \(f(x)\).
While classical percolation is a static model with the bonds or sites declared open or closed according to the values of independent Bernoulli random variables (with success probability \(p\)), in the dynamical percolation model, these Bernoulli random variables undergo a time evolution. In the simplest situation, they perform the evolution of indepedent two-state Markov chains (in equilibrium) that change value to “open” at rate \(p\) and to “closed” at rate \(1-p\).
It is well known that for critical site percolation on the triangular lattice, almost surely, there is no infinite open cluster. The main result of Schramm and Steif is that for dynamical critical site percolation on the triangular lattice, with probability one there are “exceptional times” where an infinite open cluster exists. In addition, they show that the Hausdorff dimension of the set of such times is almost surely constant and takes a value in the interval \([1/6,31/36]\). Furthermore, the authors show that there are no times where more than one infinite open cluster exists.
The results depend on the so-called critical exponents of percolation that are known explicitly for site percolation on the triangular lattice. For bond percolation on the integer lattice, even existence of these exponents has not been established. For this case, the authors have analogous but less complete results.

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C43 Time-dependent percolation in statistical mechanics
03D15 Complexity of computation (including implicit computational complexity)
42B05 Fourier series and coefficients in several variables
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