Restricted random walk

Direct evaluation of p(l, m) is not immediate. For example, in the restricted random walk on a square lattice in which squares are eliminated, the transition p 2, 2) is smaller than the transition p 3, 2), However, in an unrestricted random walk on a square lattice with no self reversals, p(2,2)=p(3,2) = i. 

The matrix Q can now be transformed into a stochastic matrix, which will be descriptive of the restricted random walks rather than of their generation employing probabilities based on unrestricted walk models. The transformation is performed as follows Let Xt be the largest eigenvalue of the matrix Q, and let Sj be the corresponding left-hand side eigenvector (defined by SjQ = X ). Let A be a diagonal matrix with elements a(i,j) = (/) 8st = [ 1(1),. v,(2),..., (v)] and 8(i,j) is the... 

Figure 12 Four subsequent STM images of a diffusing Si ad-dimer on Ge(001) at room temperature. Time lapse between the images is indicated. Image size is 16 x 16 nm2. (a) Sample bias —1.6V and tunneling current 0.4nA. (b)-(d) Sample bias +1.6 V and tunneling current 0.4 nA. The Si ad-dimer in the trough (images (a) and (b)) converts to an on-top ad-dimer (images (c) and (d)) and starts to diffuse. The diffusing ad-dimer can only perform a restricted random walk because it is trapped between two clusters separated by a distance L from each other.

Let us remind that for a usual non-restricted random walk we would obtain the well-known equation (see, for instance, Ref. [13]) ... 

A restricted random walk matrix RRW was also proposed [Randic, 1995c] as an AxA dimensional square unsymmetric matrix that enumerates restricted (i.e. selected) random walks over a molecular graph (7. The i-j entry of the matrix is the probability of a random walk starting at vertex v, and ending at vertex v,- of length equal to the topological distance dij between the considered vertices ... 

For acyclic graphs, the restricted random walk matrix is simply the reciprocal walk matrix where Mi = A, M2 = D, and M3 = 1 ... 

Randic, M. (1995c). Restricted Random Walks on Graphs. Theor.Chim.Acta, 92,97-106. 

Consider a restricted random walk on a square lattice. Let us assume that a walker is not allowed to step back (but can go forward, turn right, or turn left with equal probability). Calculate the mean-square end-to-end distance for such a restricted n-step random walk. What is the characteristic ratio Coo for this walk The lattice constant is equal to /. 

Consider a restricted random walk on a 3D cubic lattice. Let us assume That a walker is not allowed step back (but can go forward, turn up, down,... 

The restricted random-walk matrix, denoted by RRW, has been introduced by Randic (1995). The RRW matrix is defined as... 

Randic (1995) used selected invariants (paths of different lengths) of the restricted random-walk matrices for successfully deriving the structure-entropy model of octanes. 

M. Randid, Restricted random walks on graphs, Theoret. Chim. Acta 92 (1995) 97-106. 

Matrices associated with molecular graphs need not necessarily be symmetric even though the underlying molecular graph is described by a symmetric (adjacency) matrix. When one considers restricted random walks on a graph, one arrives at a non-symmetric matrix. If we restrict the number of steps in a walk by the distance between the vertices considered then in general the probability of a successful random walk from i to j is different from the probability of a successful random walk from j to /. In Table 7 we illustrate the restricted random walk matrix for graph Gi. 

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