Cumulants and Limit Theorems for q-step walks on random graphs of long-range percolation radius model

We study cumulants of q-step closed and non-closed walks on the Erd os-R\'enyi-type random graphs of long-range percolation radius model in the limit when number of vertices N, concentration c, and interaction radius R tend to infinity. These cumulants represent terms of cumulant expansion of the free energy of discrete analogs of matrix models widely known in theoretical and mathematical physics. Using a diagram technique, we show that the leading term Fₖ(q) of the normalized cumulant of k-th order can be associated with one or another family of tree-type diagrams, in dependence of the asymptotic behavior of parameters $cR/N$ for non-closed walks and c²R/N² for closed walks, respectively. Adapting the Pr\"ufer codification procedure to such diagrams, we get explicit expressions for the number of tree-type diagrams of k elements and show that in certain cases, Fₖ(q) can be bounded from above by the k-th moment of the sum of $k(q-1)+1$ independent Bernoulli random variables. These results allow us to prove Limit Theorems for the numbers of q-step walks in random graphs and to show that in the additional limiting transition of infinite q, the term Fₖ(q), considered for large values of k, admits an upper bound that asymptotically coincides with that of the k-th moment of the Compound Poisson distribution.