Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order

Abstract

Building on the inequalities for homogeneous tetrahedral polynomials in independent Gaussian variables due to R. Latała we provide a concentration inequality for not necessarily Lipschitz functions f : R n → R with bounded derivatives of higher orders, which holds when the underlying measure satisfies a family of Sobolev type inequalitiesSuch Sobolev type inequalities hold, e.g., if the underlying measure satisfies the logSobolev inequality (in which case C( p) ≤ C √ p) or the Poincaré inequality (then C( p) ≤ C p). Our concentration estimates are expressed in terms of tensor-product norms of the derivatives of f . When the underlying measure is Gaussian and f is a polynomial (not necessarily tetrahedral or homogeneous), our estimates can be reversed (up to a constant depending only on the degree of the polynomial). We also show that for polynomial functions, analogous estimates hold for arbitrary random vectors with independent sub-Gaussian coordinates. We apply our inequalities to general additive functionals of random vectors (in particular linear eigenvalue statistics of random matrices) and the problem of counting cycles of fixed length in Erdős-Rényi random graphs, obtaining new estimates, optimal in a certain range of parameters.

Subgaussian 1-cocycles on discrete groups

“…Since the gradient form Γ coincides with the modulus of the gradient if −A is the Laplacian of a Euclidean space, then (1.2) is indeed a generalization of (1.1). It is known that for classical diffusion semigroups, LSI implies (1.2); see [2] and also [1]. Efraim and Lust-Piquard proved that (1.2) holds for Walsh systems and CAR algebras in [13].…”

“…As a classical example, the Gaussian measure on R d satisfies (1.1) due to Pisier [35]; see [26] for another proof. More classical examples satisfying (1.1) can be found in [1] and the references therein. In fact, one way to generalize (1.1) is via the semigroup theory of operators.…”

A counterexample to the DeMarco‐Kahn upper tail conjecture

“…Shortly after the upper tail problem was settled for triangles H = K 3 [5,8], in 2011 DeMarco and Kahn solved the more general case of fixed-size cliques H = K r with r ⩾ 3 [9], and also formulated a plausible conjecture on the general solution of Problem 1; see Conjecture 1 below. This "upper tail conjecture" has been verified for large p = p(n) of form p ⩾ n − H via large deviation machinery [2,6,10,23], and for small p = p(n) of form p ⩽ n −v∕e (log n) C H for so-called strictly balanced graphs H [28,31,34] (where e F ∕v F < e H ∕v H for any non-empty F ⊊ H); see also [1,27,28,30] for further supporting results. In fact, this conjecture was also described as "likely to be true" in the recent random graphs book by Frieze and Karoński [12,Section 5.4].…”

“…Note that the union of G ′ and F contains at least ( |U| 2 ) ⩾ z * different 5-vertex-paths with endpoints v 3 , v 4 and internal vertices from V 2 . Recalling p 2 = n −1∕m H +o (1) , the probability of step (ii). (b) is thus at least…”

“…In 2002, Janson, Oleszkiewicz, and Ruciński [18] finally determined the exponential rate of decay (1) up to a factor of O (log(1∕p)). This breakthrough solved Problem 1 for constant edge-probabilities p ∈ (0, 1), but closing the logarithmic gap for p = o(1) remained an elusive technical challenge.…”