On some conditions for absence of a giant component in the generalised allocation scheme

Abstract

We prove two theorems which give conditions for absence of a giant component in the generalised allocation scheme and present a series of examples.

On transition of distributions of sums of independent identically distributed random variables from one lattice to another in the generalised allocation scheme

“…Some of limit theorems proved in this paper and in [6] can likely be obtained by checking the general conditions for validity of local limit theorems given in [12] (see [3]), but the authors believe that the presence of simple direct proofs may be of advantage in future studies.…”

“…But in most applications of the generalised scheme the need for local limit theorems in the triangular array scheme arises. So far, in these cases there is no complete answer whether or not a local limit theorem holds even in the case of convergence to the Gaussian law; in each particular case, one needs to carry out a separate proof, either following the pattern due to Gnedenko [1,2,11,13,14], or using the general local theorems [3,12]. Usually (see, e.g., [9]), the random variables 1 ; : : : ; N are distributed as follows:…”

On transition of distributions of sums of independent identically distributed random variables from one lattice to another in the generalised allocation scheme

“…Some of limit theorems proved in this paper and in [6] can likely be obtained by checking the general conditions for validity of local limit theorems given in [12] (see [3]), but the authors believe that the presence of simple direct proofs may be of advantage in future studies.…”

Emergence of a giant component in a random permutation with given number of cycles

“…In similar problems, the partitioning into zones is also carried out in dependence on the behaviour of the ratios n=N and ¹ 4 =.¾ 4 N/ (see, e.g., [6,7]). In similar problems, the partitioning into zones is also carried out in dependence on the behaviour of the ratios n=N and ¹ 4 =.¾ 4 N/ (see, e.g., [6,7]).…”

“…In similar problems, the partitioning into zones is also carried out in dependence on the behaviour of the ratios n=N and ¹ 4 =.¾ 4 N/ (see, e.g., [6,7]). As shown in [6,7,8], a giant component could emerge precisely under the condition N / as n !…”