We consider the full shift @math where @math , @math being a finite alphabet. For a class of potentials which contains in particular potentials @math with variation decreasing like @math for some @math , we prove that their corresponding equilibrium state @math satisfies a Gaussian concentration bound. Namely, we prove that there exists a constant @math such that, for all @math and for all separately Lipschitz functions @math , the exponential moment of @math is bounded by @math . The crucial point is that @math is independent of @math and @math . We then derive various consequences of this inequality. For instance, we obtain bounds on the fluctuations of the empirical frequency of blocks, the speed of convergence of the empirical measure, and speed of Markov approximation of @math . We also derive an almost-sure central limit theorem.
For @math is Lipschitz, Theorem was proved in @cite_7 . The main goal of @cite_7 was then to deal with nonuniformly hyperbolic systems modeled by a Young tower. For maps of the unit interval, one can have for instance maps with an indifferent fixed point. When the tower has exponential tails, the authors of @cite_7 proved a Gaussian concentration bound. When the tower has polynomial tails, they proved moment concentration bounds.
@cite_7
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For dynamical systems modeled by a Young tower with exponential tails, we prove an exponential concentration inequality for all separately Lipschitz observables of n variables. When tails are polynomial, we prove polynomial concentration inequalities. Those inequalities are optimal. We give some applications of such inequalities to specific systems and specific observables. We prove two stability results for Lipschitz functions on triangulable, compact metric spaces and consider applications of both to problems in systems biology. Given two functions, the first result is formulated in terms of the Wasserstein distance between their persistence diagrams and the second in terms of their total persistence. We prove several facts concerning Lipschitz percolation, including the following. The critical probability p_L for the existence of an open Lipschitz surface in site percolation on Z^d with d 2 satisfies the improved bound p_L 1-1 [8(d-1)]. Whenever p > p_L, the height of the lowest Lipschitz surface above the origin has an exponentially decaying tail. The lowest surface is dominated stochastically by the boundary of a union of certain independent, identically distributed random subsets of Z^d. As a consequence, for p sufficiently close to 1, the connected regions of Z^ d-1 above which the surface has height 2 or more exhibit stretched-exponential tail behaviour. There exists a Lipschitz embedding of a d-dimensional comb graph (consisting of infinitely many parallel copies of Z^ d-1 joined by a perpendicular copy) into the open set of site percolation on Z^d, whenever the parameter p is close enough to 1 or the Lipschitz constant is sufficiently large. This is proved using several new results and techniques involving stochastic domination, in contexts that include a process of independent overlapping intervals on Z, and first-passage percolation on general graphs.