ON THE CURRENT ENHANCEMENT AT THE EDGE OF A CRACK IN A LATTICE OF RESISTORS

Consider a network whose nodes are the integer lattice points and whos e arcs are fuses of 1 Omega resistance. Remove a horizontal segment of N adjacent vertical arcs, forming a 'crack' of length N. Subject the network to a uniform potential gradient of upsilon volts per are in th e north-south (or vertical) direction and measure the current in one o f the two vertical arcs at the ends of the crack. We write this curren t in the form e(N)upsilon, and call e(N) the current enhancement. We s how that the enhancement grows at a rate that is the order of the squa re root of the crack length. Our method is to identify the enhancement with the mean time to exit an interval for a certain integer valued r andom walk, and then to use some of the well-known Fourier methods for studying random walk. Our random walk has no mean or higher moments a nd is in the domain of attraction of the Cauchy law. We provide a good approximation to the enhancement using the explicitly known mean time to exit an interval for a Cauchy process. Weak convergence arguments together with an estimate of a recurrence probability enable us to sho w that the current in an intact fuse, that is in the interior of a cra ck of length N, grows proportionally with Nl log N.