arXiv subjects

Jeff Kahn

Publications and source records attributed to Jeff Kahn.

58 records · Page 4Linked to original sources

arXiv2007

Positive association in the fractional fuzzy Potts model

A fractional fuzzy Potts measure is a probability distribution on spin configurations of a finite graph $G$ obtained in two steps: first a subgraph of $G$ is chosen according to a random cluster measure $ϕ_{p,q}$, and then a spin ($\pm1$) is chosen independently for each component of the subgraph and assigned to all vertices of that component. We show that whenever $q\geq1$, such a measure is positively associated, meaning that any two increasing events are positively correlated. This generalizes earlier results of Häggström [Ann. Appl. Probab. 9 (1999) 1149--1159] and Häggström and Schramm [Stochastic Process. Appl. 96 (2001) 213--242].

math.PR↗

arXiv2006

Thresholds and expectation thresholds

Consider a random graph G in G(n,p) and the graph property: G contains a copy of a specific graph H. (Note: H depends on n; a motivating example: H is a Hamiltonian cycle.) Let q be the minimal value for which the expected number of copies of H' in G is at least 1/2 for every subgraph H' of H. Let p be the value for which the probability that G contains a copy of H is 1/2. Conjecture: p/q = O(log n). Related conjectures for general Boolean functions, and a possible connection with discrete isoperimetry are discussed.

math.CO↗

arXiv2004

Some Conditional Correlation Inequalities for Percolation and Related Processes

Consider ordinary bond percolation on a finite or countably infinite graph. Let s, t, a and b be vertices. An earlier paper proved the (nonintuitive) result that, conditioned on the event that there is no open path from s to t, the two events "there is an open path from s to a" and "there is an open path from s to b" are positively correlated. In the present paper we further investigate and generalize the theorem of which this result was a consequence. This leads to results saying, informally, that, with the above conditioning, the open cluster of s is conditionally positively (self-)associated and that it is conditionally negatively correlated with the open cluster of t. We also present analogues of some of our results for (a) random-cluster measures, and (b) directed percolation and contact processes, and observe that the latter lead to improvements of some of the results in a paper of Belitsky, Ferrari, Konno and Liggett (1997).

math.PR↗

arXiv1993

A counterexample to Borsuk's conjecture

Let $f(d)$ be the smallest number so that every set in $R^d$ of diameter 1 can be partitioned into $f(d)$ sets of diameter smaller than 1. Borsuk's conjecture was that $f(d)\! =\!d\!+\!1$. We prove that $f(d)\! \ge\! (1.2)^{\sqrt d}$ for large~$d$.

math.MG↗