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.

Researches on Pyrimidines. CXXXVI. The Mechanism of Formation of Tetrahydropyrimidines by the Biginelli Reaction1

Consider the following interacting particle system on the $d$-ary tree, known as the frog model: Initially, one particle is awake at the root and i.i.d. Poisson many particles are sleeping at every other vertex. Particles that are awake perform simple random walks, awakening any sleeping particles they encounter. We prove that there is a phase transition between transience and recurrence as the initial density of particles increases, and we give the order of the transition up to a logarithmic factor.Comment: Published at http://dx.doi.org/10.1214/15-AAP1127 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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The frog model is a growing system of random walks where a particle is added whenever a new site is visited. A longstanding open question is how often the root is visited on the infinite d-ary tree. We prove the model undergoes a phase transition, finding it recurrent for d = 2 and transient for d ≥ 5. Simulations suggest strong recurrence for d = 2, weak recurrence for d = 3, and transience for d ≥ 4. Additionally, we prove a 0-1 law for all d-ary trees, and we exhibit a graph on which a 0-1 law does not hold.To prove recurrence when d = 2, we construct a recursive distributional equation for the number of visits to the root in a smaller process and show the unique solution must be infinity a.s. The proof of transience when d = 5 relies on computer calculations for the transition probabilities of a large Markov chain. We also include the proof for d ≥ 6, which uses similar techniques but does not require computer assistance.