By Sidney Redner

Preface; Errata; 1. First-passage basics; 2. First passage in an period; three. Semi-infinite approach; four. Illustrations of first passage in basic geometries; five. Fractal and nonfractal networks; 6. platforms with round symmetry; 7. Wedge domain names; eight. purposes to uncomplicated reactions; References; Index

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The basic connection is that if all the lattice bonds are viewed as resistors (not necessarily identical), then Kirchhoff’s laws for steady current ﬂow in the + + + + + + - - V + - + - + - + (a) (b) Fig. 5. (a) A lattice graph with boundary sites B+ or B− . (b) Corresponding resistor network in which each bond (with rectangle) is a 1- resistor. The sites in B+ are all ﬁxed at potential V = 1, and sites in B− are all grounded. 34 First-Passage Fundamentals network are identical to the discrete Laplace equation for E.

The Green’s Function Formalism Let us recall the conventional method to compute the ﬁrst-passage probability of diffusion. Consider a diffusing particle that starts at r 0 within a domain with boundary B. 1) with the initial condition c(r , 0; r 0 ) = δ(r − r 0 ) and the absorbing boundary condition c(r , t; r 0 )|r ∈B = 0. This condition accounts for the fact that once a particle reaches the boundary it leaves the system. Because the initial condition is invariably a single particle at r 0 , we will typically not write this argument in the Green’s function.

Because the voltage V j also equals the probability for the corresponding random walk to reach B+ without reaching B− , the term V j P+ j is just the probability that a random walk starts at B+ , makes a single step to the sites j (with hopping probabilities Pi j ), and then returns to B+ without reaching B− . We therefore deduce that I = (1 − V j )P+ j g+ j j = j g+ j × (1 − return probability) j g+ j × escape probability. 4) j Here “escape” means reaching the opposite terminal of the voltage source without returning to the starting point.