Size biased couplings and the spectral gap for random regular graphs

Abstract

Let λ be the second largest eigenvalue in absolute value of a uniform random dregular graph on n vertices. It was famously conjectured by Alon and proved by Friedman that if d is fixed independent of n, then λ = 2 √ d − 1 + o(1) with high probability. In the present work we show that λ = O( √ d) continues to hold with high probability as long as d = O(n 2/3 ), making progress towards a conjecture of Vu that the bound holds for all 1 ≤ d ≤ n/2. Prior to this work the best result was obtained by Broder, Frieze, Suen and Upfal (1999) using the configuration model, which hits a barrier at d = o(n 1/2 ). We are able to go beyond this barrier by proving concentration of measure results directly for the uniform distribution on d-regular graphs. These come as consequences of advances we make in the theory of concentration by size biased couplings. Specifically, we obtain Bennett-type tail estimates for random variables admitting certain unbounded size biased couplings.

Relaxation of monotone coupling conditions: Poisson approximation and beyond

“…As we will see, the upper bounds we obtain have a form similar to that of (2). The condition (8) below is employed directly by [7], where a random variable Y satisfying this condition is said to be '1-bounded with probability p for the upper tail'. The analogous condition (10) was not used by [7], but is in the same spirit.…”

“…In Section 2 we derive Poisson approximation bounds (Theorem 2.2) for random variables W which come close (in a certain sense) to satisfying either (3) or its positive dependence analogue. This relaxation of these conditions is motivated by recent work of Cook, Goldstein and Johnson [7], who established a concentration inequality which can control the spectral gap of random graphs. In Section 2.2 we use Theorem 2.2 to derive explicit Poisson approximation results for models where an underlying random variable of interest W , satisfying (3), or its positive dependence analogue, is contaminated by noise.…”

Structure of eigenvectors of random regular digraphs

“…The following statement is a non-Hermitian counterpart of the spectral gap estimates for undirected random d-regular graphs -a characteristic of major importance in connection with the graph expansion properties. We refer to [1,10,13,17,19,21,24,25,32,46,57] for more information on random expanders. The following statement was first proved in [13] for d ≤ C √ n (which is enough in this paper), then it was extended to the range d ≤ Cn 2/3 in [17] and to d ≤ n/2 in [57].…”

“…Denote h := h(L). By the definition of H b , the construction of the ℓ-decomposition, (17), and (18), we have…”

Finite plateau in spectral gap of polychromatic constrained random networks

“…The degree of the degeneration of the Laplacian mode, λ 1 , defines the number of disconnected components of the graph. The behavior of the second eigenvalue of the Laplacian, λ 2 , in RRG is the subject of the several mathematical studies [5]. It has an important meaning for any network known as "the algebraic connectivity".…”