207 research outputs found

### 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

### 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

### Rotor walks on general trees

The rotor walk on a graph is a deterministic analogue of random walk. Each
vertex is equipped with a rotor, which routes the walker to the neighbouring
vertices in a fixed cyclic order on successive visits. We consider rotor walk
on an infinite rooted tree, restarted from the root after each escape to
infinity. We prove that the limiting proportion of escapes to infinity equals
the escape probability for random walk, provided only finitely many rotors send
the walker initially towards the root. For i.i.d. random initial rotor
directions on a regular tree, the limiting proportion of escapes is either zero
or the random walk escape probability, and undergoes a discontinuous phase
transition between the two as the distribution is varied. In the critical case
there are no escapes, but the walker's maximum distance from the root grows
doubly exponentially with the number of visits to the root. We also prove that
there exist trees of bounded degree for which the proportion of escapes
eventually exceeds the escape probability by arbitrarily large o(1) functions.
No larger discrepancy is possible, while for regular trees the discrepancy is
at most logarithmic.Comment: 32 page

- …