Correlations for paths in random orientations of G(n,p) and G(n,m)

We study random graphs, both $G(n,p)$ and $G(n,m)$, with random orientations on the edges. For three fixed distinct vertices s,a,b we study the correlation, in the combined probability space, of the events a -> s and s -> b. For G(n,p), we prove that there is a p_c=1/2 such that for a fixed p<p_c the correlation is negative for large enough n and for p>p_c the correlation is positive for large enough n. We conjecture that for a fixed n\ge 27 the correlation changes sign three times for three critical values of p. For G(n,m) it is similarly proved that, with $p=m/\binom{n}{2}$, there is a critical p_c that is the solution to a certain equation and approximately equal to 0.7993. A lemma, which computes the probability of non existence of any k directed edges in G(n,m), is thought to be of independent interest. We present exact recursions to compute P(a -> s)$and P(a -> s, s -> b)$. We also briefly discuss the corresponding question in the quenched version of the problem.Comment: Author added, main proof greatly simplified and extended to cover also G(n,m). Discussion on quenched version adde

Monotonicity of the difference between median and mean of Gamma distributions and of a related Ramanujan sequence

For n # 0, let # n be the median of the #(n + 1, 1) distribution. We prove that the sequence {# n = # n - n} decreases from log 2 to 2/3 as n increases from 0 to #

Upper and Lower Bounds for the Connective Constants of Self-avoiding Walks on the Archimedean and Laves Lattices

We give improved upper and lower bounds for the connective constants of self-avoiding walks on a class of lattices, including the Archimedean and Laves lattices. The lower bounds are obtained by using Kesten’s method of irreducible bridges, with an appropriate generalisation for weakly regular lattices. The upper bounds are obtained as the largest eigenvalue of a certain transfer matrix. The obtained bounds show that, in the studied class of lattices, the connective constant is increasing in the average degree of the lattice. We also discuss an alternative measure of average degree