Sven Erick Alm

Upper Bounds for the Connective Constant of Self-Avoiding Walks

We present a method for obtaining upper bounds for the connective constant of selfavoiding walks. The method works for a large class of lattices, including all that have been studied in connection with self-avoiding walks. The bound is obtained as the largest eigenvalue of a certain matrix. Numerical application of the method has given improved bounds for all lattices studied, e.g. ji < 2.696 for the square lattice, n < 4.278 for the triangular lattice and n < 4.756 for the simple cubic lattice.

Approximation and Simulation of the Distributions of Scan Statistics for Poisson Processes in Higher Dimensions

Given a Poisson process in two or three dimensions, we are interested in the scan statistic, i.e. the largest number of points contained in a translate of a fixed scanning set restricted to lie inside a rectangular area. The distribution of the scan statistic is accurately approximated for rectangular scanning sets, using a technique that is also extended to higher dimensions. The accuracy of the approximation is checked through simulation.

Exact Expectations and Distributions for the Random Assignment Problem

A generalization of the random assignment problem asks the expected cost of the minimum-cost matching of cardinality k in a complete bipartite graph Km,n, with independent random edge weights. With weights drawn from the exponential distribution with intensity 1, the answer has been conjectured to beΣi,j≥0, i+j<k1/(m−i)(n−j).Here, we prove the conjecture for k l 4, k = m = 5, and k = m = n = 6, using a structured, automated proof technique that results in proofs with relatively few cases. The method yields not only the minimum assignment costs expectation but the Laplace transform of its distribution as well. From the Laplace transform we compute the variance in these cases, and conjecture that, with k = m = n → ∞, the variance is 2/n + O(log n/n2). We also include some asymptotic properties of the expectation and variance when k is fixed.

Approximations of the Distributions of Scan Statistics of Poisson Processes

We study scan statistics of Poisson processes in one and higher dimensions. First, a very accurate approximation is established in one dimension. This is done by studying uperossings of the scanning process and noting that these occur in clusters. The clusters appear more or less independently and the cluster size is estimated by a random walk argument. This idea is then used repeatedly to obtain approximations in higher dimensions. Simulation is used to check the accuracy of the approximations in two and three dimensions. A discussion of these simulations is included, as they are by no means trivial to perform.

Inequalities for Means of Restricted First-Passage Times in Percolation Theory

A simple geometric argument establishes an inequality between the sums of two pairs of first-passage times. This result is used to prove monotonicity, convexity and concavity results for first-passage times with cylinder and half-space restrictions.

Lower and Upper Bounds for the Time Constant of First-Passage Percolation

We present improved lower and upper bounds for the time constant of first-passage percolation on the square lattice. For the case of lower bounds, a new method, using the idea of a transition matrix, has been used. Numerical results for the exponential and uniform distributions are presented. A simulation study is included, which results in new estimates and improved upper confidence limits for the time constants.

Bounds for the connective constant of the hexagonal lattice

We give improved bounds for the connective constant of the hexagonal lattice. The lower bound is found by using Kestens method of irreducible bridges and by determining generating functions for br ...

Upper and lower bounds for the connective constants of self-avoiding walks on the Archimedean and Laves lattices

We give improved upper and lower bounds for the connective constants of self-avoiding walks on a class of lattices, including the Archimedean and Laves lattices. The lower bounds are obtained by using Kestens method of irreducible bridges, with an appropriate generalization for weakly regular lattices. The upper bounds are obtained as the largest eigenvalue of a certain transfer matrix. The obtained bounds show that, in the studied class of lattices, the connective constant is increasing in the average degree of the lattice. We also discuss an alternative measure of average degree.

A Note on a Problem by Welsh in First-Passage Percolation

Consider first-passage percolation on the square lattice. Welsh, who together with Hammersley introduced the subject in 1963, has formulated a problem about mean first-passage times, which, although seemingly simple, has not been proved in any non-trivial case. In this paper we give a general proof of Welshs problem.

A counter-intuitive correlation in a random tournament

Consider a randomly oriented graph G = (V, E) and let a, s and b be three distinct vertices in V. We study the correlation between the events {a → s} and {s → b}. We show that, counter-intuitively, when G is the complete graph Kn, n ≥ 5, then the correlation is positive. (It is negative for n = 3 and zero for n = 4.) We briefly discuss and pose problems for the same question on other graphs.