Finite size corrections to disordered Ising models on Random Regular   Graphs
  We derive the analytical expression for the first finite size correction to the average free energy of disordered Ising models on random regular graphs. The formula can be physically interpreted as a weighted sum over all non self-intersecting loops in the graph, the weight being the free-energy shift due to the addition of the loop to an infinite tree. 
Carlo Lucibello, Flaviano Morone, Giorgio Parisi, Federico   Ricci-Tersenghi, Tommaso Rizzo
cond-mat.dis-nn