We present a method for obtaining upper bounds for the connective constant of self-avoiding 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. μ < 2.696 for the square lattice, μ < 4.278 for the triangular lattice and μ < 4.756 for the simple cubic lattice.
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 [les ] 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 cost's 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.
We give improved bounds for the connective constant of the hexagonal lattice. The lower bound is found by using Kesten's method of irreducible bridges, and by determining generating functions for bridges on one-dimensional lattices. The upper bound is obtained as the largest eigenvalue of a certain transfer matrix. Using a relation between the hexagonal and the (3.12 2) lattices, we also give bounds for the connective constant of the latter lattice.
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.
Given a two-dimensional Poisson process, X, with intensity λ, we are interested in the largest number of points, L, contained in a translate of a fixed scanning set, C, restricted to lie inside a rectangular area.
The distribution of L is accurately approximated for rectangular scanning sets, using a technique that can be extended to higher dimensions. Reasonable approximations for non-rectangular scanning sets are also obtained using a simple correction of the rectangular result.
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