Network on Chip (NoC) is an enabling methodology of integrating a very high number of intellectual property (IP) blocks in a single System on Chip (SoC). A major challenge that NoC design is expected to face is the intrinsic unreliability of the interconnect infrastructure under technology limitations. Research must address the combination of new device-level defects or error-prone technologies...

We report on a Monte Carlo study of so-called two-choice-spiral self-avoiding walks on the square lattice. These have the property that their geometric size (such as is measured by the radius of gyration) scales anisotropically, with exponent values that seem to defy rational fraction conjectures. This polymer model was previously understood to be in a universality class different to ordinary s...

The number of even 321-avoiding permutations of length n is equal to the number of odd ones if n is even, and exceeds it by the n−1 2 th Catalan number otherwise. We present an involution that proves a refinement of this sign-balance property respecting the length of the longest increasing subsequence of the permutation. In addition, this yields a combinatorial proof of a recent analogous resul...

Let µ be the self-avoiding walk connective constant on Z d. We show that the asymptotic expansion for β c = 1/µ in powers of 1/(2d) satisfies Borel type bounds. This supports the conjecture that the expansion is Borel summable.

We consider a model of self-avoiding walks on the lattice Z with different weights for steps in each of the 2d lattice directions. We find that the directiondependent mass for the two-point function of this model has three phases: mass positive in all directions; mass identically −∞; and masses of different signs in different directions. The final possibility can only occur if the weights are a...

The partition function of the O(n) loop model on the honeycomb lattice is mapped to that of the O(n) loop model on the 3-12 lattice. Both models share the same operator content and thus critical exponents. The critical points are related via a simple transformation of variables. When n = 0 this gives the recently found exact value μ = 1.711 041 . . . for the connective constant of self-avoiding...

In [BEI] we introduced a Levy process on a hierarchical lattice which is four dimensional, in the sense that the Green’s function for the process equals 1 |x|2 . If the process is modified so as to be weakly self-repelling, it was shown that at the critical killing rate (mass-squared) βc, the Green’s function behaves like the free one. Now we analyze the end-to-end distance of the model and sho...

Using a novel implementation of the Goulden-Jackson method, we compute new upper bounds for the connective constants of self-avoiding walks, breaking Alm's previous records for rectangular (hypercubic) lattices. We also give the explicit generating functions for memory 8. The new upper bounds are 2.

Abstract: We give a survey and unified treatment of functional integral representations for both simple random walk and some self-avoiding walk models, including models with strict self-avoidance, with weak selfavoidance, and a model of walks and loops. Our representation for the strictly self-avoiding walk is new. The representations have recently been used as the point of departure for rigoro...

This is the second of two papers on the end-to-end distance of a weakly selfrepelling walk on a four dimensional hierarchical lattice. It completes the proof that the expected value grows as a constant times √ T log 1 8 T (