Statistics of SAWs on a -dimensional lattice

In accordance with the standard procedure [2] (see also [3]) of Gaussian distribution in the deduction of RW statistics, we will take into account that the one-dimensional chain-end walk has a step equal to root-mean-square chain link length projection on the walking direction, and the number of steps for each direction is equal to the length of chain N. So, each step of walk is actually t/-dimensional, and transition to r/-dimensional distribution function is unnecessary. 

Let us incorporate the numbers n, of chain end steps in the i direction of the d-dimensional lattice with cell size equal to the statistical length of the chain link  

The quantity of realization variants of random steps in i directions is equal to ni. /n nf , where the numbers of steps in positive (n ) and negative (n,) directions of wandering are correlated by n +nr=tii. Considering that the probability of wandering in the positive or negative direction chosen is assumed to be the same and equal to 1/2, the 

incorporating the quantity of effective steps in i directions of wandering Si-n -nf, we obtain n - n-, + S/)/2, -i,)/2. Then (8.6) can look as follows  

One can see that the change of sign at s, does not change the value coin). Thus, fflfn) represents the probability that the trajectory of random wandering after n,- steps in i directions will end in one of 2 cells Mp s), coordinates of which are determined by vectors j= (s ), (=, d having a distinction in signs of their components s, only. These cells or states of chain end are equiprobable. 

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