4 research outputs found
A numerical adaptation of SAW identities from the honeycomb to other 2D lattices
Recently, Duminil-Copin and Smirnov proved a long-standing conjecture by
Nienhuis that the connective constant of self-avoiding walks on the honeycomb
lattice is A key identity used in that proof depends on
the existence of a parafermionic observable for self-avoiding walks on the
honeycomb lattice. Despite the absence of a corresponding observable for SAW on
the square and triangular lattices, we show that in the limit of large
lattices, some of the consequences observed on the honeycomb lattice persist on
other lattices. This permits the accurate estimation, though not an exact
evaluation, of certain critical amplitudes, as well as critical points, for
these lattices. For the honeycomb lattice an exact amplitude for loops is
proved.Comment: 21 pages, 7 figures. Changes in v2: Improved numerical analysis,
giving greater precision. Explanation of why we observe what we do. Extra
reference
Two-dimensional self-avoiding walks and polymer adsorption: Critical fugacity estimates
Recently Beaton, de Gier and Guttmann proved a conjecture of Batchelor and
Yung that the critical fugacity of self-avoiding walks interacting with
(alternate) sites on the surface of the honeycomb lattice is . A
key identity used in that proof depends on the existence of a parafermionic
observable for self-avoiding walks interacting with a surface on the honeycomb
lattice. Despite the absence of a corresponding observable for SAW on the
square and triangular lattices, we show that in the limit of large lattices,
some of the consequences observed for the honeycomb lattice persist
irrespective of lattice. This permits the accurate estimation of the critical
fugacity for the corresponding problem for the square and triangular lattices.
We consider both edge and site weighting, and results of unprecedented
precision are achieved. We also \emph{prove} the corresponding result fo the
edge-weighted case for the honeycomb lattice.Comment: 12 pages, 3 figures, 7 table
Integrability as a consequence of discrete holomorphicity: the Z_N model
It has recently been established that imposing the condition of discrete
holomorphicity on a lattice parafermionic observable leads to the critical
Boltzmann weights in a number of lattice models. Remarkably, the solutions of
these linear equations also solve the Yang-Baxter equations. We extend this
analysis for the Z_N model by explicitly considering the condition of discrete
holomorphicity on two and three adjacent rhombi. For two rhombi this leads to a
quadratic equation in the Boltzmann weights and for three rhombi a cubic
equation. The two-rhombus equation implies the inversion relations. The
star-triangle relation follows from the three-rhombus equation. We also show
that these weights are self-dual as a consequence of discrete holomorphicity.Comment: 11 pages, 7 figures, some clarifications and a reference adde