The problem of optimizing over random structures emerges in many areas of science and
engineering, ranging from statistical physics to machine learning and artificial intelligence …

We introduce a new approach to proving that a sequence of deterministic linear codes
achieves capacity on an erasure channel under maximum a posteriori decoding. Rather …

Community detection in graphs is the problem of finding groups of vertices which are more
densely connected than they are to the rest of the graph. This problem has a long history, but …

Vindicating a sophisticated but non-rigorous physics approach called the cavity method, we
establish a formula for the mutual information in statistical inference problems induced by …

Let Φ be a uniformly random k-SAT formula with n variables and m clauses. We study the
algorithmic task of finding a satisfying assignment of Φ. It is known that satisfying …

Quantum algorithms can deliver asymptotic speedups over their classical counterparts.
However, there are few cases where a substantial quantum speedup has been worked out …

Let P:{0, 1} k→{0, 1} be a nontrivial k-ary predicate. Consider a random instance of the
constraint satisfaction problem (P) on n variables with Δ n constraints, each being P applied …

Providing a graduate-level introduction to discrete probability and its applications, this book
develops a toolkit of essential techniques for analysing stochastic processes on graphs …

We analyze the matrix factorization problem. Given a noisy measurement of a product of two
matrices, the problem is to estimate back the original matrices. It arises in many applications …

Since the early 2000s physicists have developed an ingenious but non-rigorous formalism
called the cavity method to put forward precise conjectures as to the phase transitions in …