A sender holds a word x consisting of n blocks x i , each of t bits, and wishes to broadcast a codeword to m receivers, R 1 , ..., R m . Each receiver R i is interested in one block, and has prior side information consisting of some subset of the other blocks. Let β t be the minimum number of bits that has to be transmitted when each block is of length t, and let β be the limit β = lim t→∞ β t /t. In words, β is the average communication cost per bit in each block (for long blocks). Finding the coding rate β, for such an informed broadcast setting, generalizes several coding theoretic parameters related to Informed Source Coding on Demand, Index Coding and Network Coding.In this work we show that usage of large data blocks may strictly improve upon the trivial encoding which treats each bit in the block independently. To this end, we provide general bounds on β t , and prove that for any constant C there is an explicit broadcast setting in which β = 2 but β 1 > C. One of these examples answers a question of [15].In addition, we provide examples with the following counterintuitive direct-sum phenomena. Consider a union of several mutually independent broadcast settings. The optimal code for the combined setting may yield a significant saving in communication over concatenating optimal encodings for the individual settings. This result also provides new non-linear coding schemes which improve upon the largest known gap between linear and non-linear Network Coding, thus improving the results of [8].The proofs are based on a relation between this problem and results in the study of Witsenhausen's rate, OR graph products, colorings of Cayley graphs, and the chromatic numbers of Kneser graphs.

We compute the mixing rate of a non-backtracking random walk on a regular expander. Using some properties of Chebyshev polynomials of the second kind, we show that this rate may be up to twice as fast as the mixing rate of the simple random walk. The closer the expander is to a Ramanujan graph, the higher the ratio between the above two mixing rates is.As an application, we show that if G is a high-girth regular expander on n vertices, then a typical non-backtracking random walk of length n on G does not visit a vertex more than (1 + o (1)) log n log log n times, and this result is tight. In this sense, the multi-set of visited vertices is analogous to the result of throwing n balls to n bins uniformly, in contrast to the simple random walk on G, which almost surely visits some vertex Ω(log n) times.

Abstract. The following question is due to Chatterjee and Varadhan (2011). Fix 0 < p < r < 1 and take G ∼ G(n, p), the Erdős-Rényi random graph with edge density p, conditioned to have at least as many triangles as the typical G(n, r). Is G close in cut-distance to a typical G(n, r)? Via a beautiful new framework for large deviation principles in G(n, p), Chatterjee and Varadhan gave bounds on the replica symmetric phase, the region of (p, r) where the answer is positive. They further showed that for any small enough p there are at least two phase transitions as r varies.We settle this question by identifying the replica symmetric phase for triangles and more generally for any fixed d-regular graph. By analyzing the variational problem arising from the framework of Chatterjee and Varadhan we show that the replica symmetry phase consists of all (p, r) such that (r d , hp(r)) lies on the convex minorant of x → hp(x 1/d ) where hp is the rate function of a binomial with parameter p. In particular, the answer for triangles involves hp( √ x) rather than the natural guess of hp(x 1/3 ) where symmetry was previously known. Analogous results are obtained for linear hypergraphs as well as the setting where the largest eigenvalue of G ∼ G(n, p) is conditioned to exceed the typical value of the largest eigenvalue of G(n, r). Building on the work of Chatterjee and Diaconis (2012) we obtain additional results on a class of exponential random graphs including a new range of parameters where symmetry breaking occurs. En route we give a short alternative proof of a graph homomorphism inequality due to Kahn (2001) and Galvin and Tetali (2004).

Abstract. What is the probability that the number of triangles in Gn,p, the Erdős-Rényi random graph with edge density p, is at least twice its mean? Writing it as exp[−r(n, p)], already the order of the rate function r(n, p) was a longstanding open problem when p = o(1), finally settled in 2012 by Chatterjee and by DeMarco and Kahn, who independently showed