Partition Identities and the Coin Exchange Problem

The number of partitions of n into parts divisible by a or b equals the number of partitions of n in which each part and each difference of two parts is expressible as a non-negative integer combination of a or b. This generalizes identities of MacMahon and Andrews. The analogous identities for three or more integers (in place of a,b) hold in certain cases.Comment: 6 page

Slow Convergence in Bootstrap Percolation

In the bootstrap percolation model, sites in an L by L square are initially infected independently with probability p. At subsequent steps, a healthy site becomes infected if it has at least 2 infected neighbours. As (L,p)->(infinity,0), the probability that the entire square is eventually infected is known to undergo a phase transition in the parameter p log L, occurring asymptotically at lambda = pi^2/18. We prove that the discrepancy between the critical parameter and its limit lambda is at least Omega((log L)^(-1/2)). In contrast, the critical window has width only Theta((log L)^(-1)). For the so-called modified model, we prove rigorous explicit bounds which imply for example that the relative discrepancy is at least 1% even when L = 10^3000. Our results shed some light on the observed differences between simulations and rigorous asymptotics.Comment: 22 pages, 3 figure

Insertion and deletion tolerance of point processes

We develop a theory of insertion and deletion tolerance for point processes. A process is insertion-tolerant if adding a suitably chosen random point results in a point process that is absolutely continuous in law with respect to the original process. This condition and the related notion of deletion-tolerance are extensions of the so-called finite energy condition for discrete random processes. We prove several equivalent formulations of each condition, including versions involving Palm processes. Certain other seemingly natural variants of the conditions turn out not to be equivalent. We illustrate the concepts in the context of a number of examples, including Gaussian zero processes and randomly perturbed lattices, and we provide applications to continuum percolation and stable matching

Which way do I go? Neural activation in response to feedback and spatial processing in a virtual T-maze

In 2 human event-related brain potential (ERP) experiments, we examined the feedback error-related negativity (fERN), an ERP component associated with reward processing by the midbrain dopamine system, and the N170, an ERP component thought to be generated by the medial temporal lobe (MTL), to investigate the contributions of these neural systems toward learning to find rewards in a "virtual T-maze" environment. We found that feedback indicating the absence versus presence of a reward differentially modulated fERN amplitude, but only when the outcome was not predicted by an earlier stimulus. By contrast, when a cue predicted the reward outcome, then the predictive cue (and not the feedback) differentially modulated fERN amplitude. We further found that the spatial location of the feedback stimuli elicited a large N170 at electrode sites sensitive to right MTL activation and that the latency of this component was sensitive to the spatial location of the reward, occurring slightly earlier for rewards following a right versus left turn in the maze. Taken together, these results confirm a fundamental prediction of a dopamine theory of the fERN and suggest that the dopamine and MTL systems may interact in navigational learning tasks

Stochastic Domination and Comb Percolation

There exists a Lipschitz embedding of a d-dimensional comb graph (consisting of infinitely many parallel copies of Z^{d-1} joined by a perpendicular copy) into the open set of site percolation on Z^d, whenever the parameter p is close enough to 1 or the Lipschitz constant is sufficiently large. This is proved using several new results and techniques involving stochastic domination, in contexts that include a process of independent overlapping intervals on Z, and first-passage percolation on general graphs.Comment: 21 page