SIZE BIASED COUPLINGS AND THE SPECTRAL GAP FOR RANDOM REGULAR GRAPHS

Let lambda be the second largest eigenvalue in absolute value of a uniform random d-regular graph on n vertices. It was famously conjectured by Alon and proved by Friedman that if d is fixed independent of n, then lambda = 2 root d - 1+ o(1) with high probability. In the present work, we show that lambda = O(root d) continues to hold with high probability as long as d = O(n(2/3)), making progress toward 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.