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.