The paper is devoted to the problem of establishing right-convergence of
sparse random graphs. This concerns the convergence of the logarithm of number
of homomorphisms from graphs or hyper-graphs \G_N, N\ge 1 to some target
graph W. The theory of dense graph convergence, including random dense
graphs, is now well understood, but its counterpart for sparse random graphs
presents some fundamental difficulties. Phrased in the statistical physics
terminology, the issue is the existence of the log-partition function limits,
also known as free energy limits, appropriately normalized for the Gibbs
distribution associated with W. In this paper we prove that the sequence of
sparse \ER graphs is right-converging when the tensor product associated with
the target graph W satisfies certain convexity property. We treat the case of
discrete and continuous target graphs W. The latter case allows us to prove a
special case of Talagrand's recent conjecture (more accurately stated as level
III Research Problem 6.7.2 in his recent book), concerning the existence of the
limit of the measure of a set obtained from RN by intersecting it with
linearly in N many subsets, generated according to some common probability
law.
Our proof is based on the interpolation technique, introduced first by Guerra
and Toninelli and developed further in a series of papers. Specifically, Bayati
et al establish the right-convergence property for Erdos-Renyi graphs for some
special cases of W. In this paper most of the results in this paper follow as
a special case of our main theorem.Comment: 22 page
