Consider a network whose nodes are the integer lattice points and whose arcs are fuses of 1Ω resistance. Remove a horizontal segment ofNadjacent vertical arcs, forming a ‘crack’ of lengthN. Subject the network to a uniform potential gradient ofvvolts per arc in the north-south (or vertical) direction and measure the current in one of the two vertical arcs at the ends of the crack. We write this current in the forme(N)v, and calle(N) thecurrent enhancement.We show that the enhancement grows at a rate that is the order of the square 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 random 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 and 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 show that the current in an intact fuse, that is in the interior of a crack of lengthN, grows p roportionally withN/logN.
Heat and work distributions for mixed Gauss–Cauchy process
