Related papers: Self-avoiding polygons on the square lattice
We compute the exponent gamma for self-avoiding walks in three dimensions. We get gamma = 1.1575 +- 0.0006 in agreement with renormalization-group predictions. Earlier Monte Carlo and exact-enumeration determinations are now seen to be…
We present a high-statistics Monte Carlo determination of the exponent gamma for self-avoiding walks on a Manhattan lattice in two dimensions. A conservative estimate is $\gamma \gtapprox 1.3425(3)$, in agreement with the universal value…
This is a rather personal review of the problem of self-avoiding walks and polygons. After defining the problem, and outlining what is known rigorously and what is merely conjectured, I highlight the major outstanding problems. I then give…
Theta-graphs are a type of spatial graph with two vertices connected by three edges. We investigate embeddings of theta-graphs in the square and simple cubic lattices, using a combination of the Wang-Landau Monte Carlo method with a variant…
The phase diagram and surface critical behaviour of the vertex-interacting self-avoiding walk are examined using transfer matrix methods extended using DMRG and coupled with finite-size scaling. Particular attention is paid to the critical…
We present the formula for the number of monic irreducible polynomials of degree $n$ over the finite field $\mathbb F_q$ where the coefficients of $x^{n-1}$ and $x$ vanish for $n\ge3$. In particular, we give a relation between rational…
We propose a novel method to locate the QCD critical point by constructing an expansion along contours of constant entropy density. Applying two independent analysis of lattice QCD data at zero baryon chemical potential, we find a critical…
We calculate the correction-to-scaling exponent $\omega_T$ that characterizes the approach to the scaling limit in multicomponent polymer solutions. A direct Monte Carlo determination of $\omega_T$ in a system of interacting self-avoiding…
Spectra of suitably chosen Pisot-Vijayaraghavan numbers represent non-trivial examples of self-similar Delone point sets of finite local complexity, indispensable in quasicrystal modeling. For the case of quadratic Pisot units we…
We establish that the extension complexity of the nXn correlation polytope is at least 1.5^n by a short proof that is self-contained except for using the fact that every face of a polyhedron is the intersection of all facets it is contained…
Among integral polytopes (vertices with integral coordinates), lattice-free polytopes - intersecting the lattice ONLY at their vertices- are of particular interestin combinatorics and geometry of numbers. A natural question is to measure…
We present a detailed numerical study of the orthogonality catastrophe exponent for a one-dimensional lattice model of spinless fermions with nearest neighbor interaction using the density matrix remormalization group algorithm. Keeping up…
In this note, we establish an asymptotic formula for the number of rational points of bounded height on the singular cubic surface $$ x_0(x_1^2 + x_2^2)=x_3^3 $$ with a power-saving error term, which verifies the Manin-Peyre conjectures for…
Linearized polynomials appear in many different contexts, such as rank metric codes, cryptography and linear sets, and the main issue regards the characterization of the number of roots from their coefficients. Results of this type have…
We give an improved algorithm for counting the number of $1324$-avoiding permutations, resulting in 5 further terms of the generating function. We analyse the known coefficients and find compelling evidence that unlike other classical…
It is shown that the resistance between the origin and any lattice point (l,m,n) in an infinite perfect Simple Cubic (SC) is expressed rationally in terms of the known value of G0(0,0,0). The resistance between arbitrary sites in a SC is…
We study terminally attached self-avoiding walks and bridges on the simple cubic lattice, both by series analysis and Monte Carlo methods. We provide strong numerical evidence supporting a scaling relation between self-avoiding walks,…
We investigate the knotting probability after a local strand passage is performed in an unknotted self-avoiding polygon on the simple cubic lattice. We assume that two polygon segments have already been brought close together for the…
Let $c_n = c_n(d)$ denote the number of self-avoiding walks of length $n$ starting at the origin in the Euclidean nearest-neighbour lattice $\mathbb{Z}^d$. Let $\mu = \lim_n c_n^{1/n}$ denote the connective constant of $\mathbb{Z}^d$. In…
We have derived long series expansions of the percolation probability for site, bond and site-bond percolation on the directed triangular lattice. For the bond problem we have extended the series from order 12 to 51 and for the site problem…