Will Perkins

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The problem of identifying a planted assignment given a random k-SAT formula consistent with the assignment exhibits a large algorithmic gap: while the planted solution can always be identified given a formula with O(n log n) clauses, there are distributions over clauses for which the best known efficient algorithms require n<sup>k/2</sup> clauses. We(More)
The Erdős-Rényi process begins with an empty graph on n vertices, with edges added randomly one at a time to the graph. A classical result of Erdős and Rényi states that the Erdős-Rényi process undergoes a phase transition, which takes place when the number of edges reaches n/2 (we say at time 1) and a giant component emerges. Since this seminal work of(More)
The factors that determine development of single and multiple primary cutaneous basal cell carcinomas (BCCs) are unclear. We describe a case-control study firstly, to examine the influence of allelism at the glutathione S-transferase GSTM1 and GSTT1 and cytochrome P450 CYP2D6 loci on susceptibility to these tumours and, secondly, to identify interactions(More)
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 random graphs. This general result implies the conjecture on the information-theoretic threshold in the disassortative stochastic block model [Decelle et al.: Phys. Rev. E(More)
We study an Achlioptas-process version of the random k-SAT process: a bounded number of k-clauses are drawn uniformly at random at each step, and exactly one added to the growing formula according to a particular rule. We prove the existence of a rule that shifts the satisfiability threshold. This extends a well-studied area of probabilistic combinatorics(More)
Thirty-seven HIV-infected homosexual men with thrombocytopenia (less than 100 x 10(9)/l) received protein A immunoadsorption treatments to remove platelet-sensitizing immunoglobulin (Ig) G and circulating immune complexes (CIC) from plasma. Patients received an average of six treatments each, consisting of 250 ml plasma over a 3-week period. Clinical(More)
1 We consider a bipartite stochastic block model on vertex sets V1 and V2, with planted partitions in each, and ask at what densities efficient algorithms can recover the partition of the smaller vertex set. When |V2| |V1|, multiple thresholds emerge. We first locate a sharp threshold for detection of the partition, in the sense of the results of Mossel et(More)
According to physics predictions, the free energy of random factor graph models that satisfy a certain “static replica symmetry” condition can be calculated via the Belief Propagation message passing scheme [20]. Here we prove this conjecture for a wide class of random factor graph models. Specifically, we show that the messages constructed just as in the(More)