The logarithmic Sobolev constant is always bounded above by half the spectral gap. It is natural to ask when this inequality is an equality. We consider this question in the context of reversible Markov chains on small finite state spaces. In particular, we prove that equality holds for simple random walk on the five cycle and we discuss assorted families of chains on three and four points.
We revisit the problem of selecting an item from n choices that appear before us in random sequential order so as to minimize the expected rank of the item selected. In particular, we examine the stopping rule where we reject the first k items and then select the first subsequent item that ranks lower than the l-th lowest-ranked item among the first k. We prove that the optimal rule has k ~ n/e, as in the classical secretary problem where our sole objective is to select the item of lowest rank;...
We prove that the uniform infinite random quadrangulations defined respectively by Chassaing–Durhuus and Krikun have the same distribution.
Let Gₙ be the random graph on [n] = 1,...,n with the possible edge i,j having probability for j ≠ i, i+1, i-1 with α ∈ (0,1) irrational. We prove that the zero-one law (for first order logic) holds..