7,306 research outputs found
Equality of bond percolation critical exponents for pairs of dual lattices
For a certain class of two-dimensional lattices, lattice-dual pairs are shown
to have the same bond percolation critical exponents. A computational proof is
given for the martini lattice and its dual to illustrate the method. The result
is generalized to a class of lattices that allows the equality of bond
percolation critical exponents for lattice-dual pairs to be concluded without
performing the computations. The proof uses the substitution method, which
involves stochastic ordering of probability measures on partially ordered sets.
As a consequence, there is an infinite collection of infinite sets of
two-dimensional lattices, such that all lattices in a set have the same
critical exponents.Comment: 10 pages, 7 figure
On entropy, specific heat, susceptibility and Rushbrooke inequality in percolation
We investigate percolation, a probabilistic model for continuous phase
transition (CPT), on square and weighted planar stochastic lattices. In its
thermal counterpart, entropy is minimally low where order parameter (OP) is
maximally high and vice versa. Besides, specific heat, OP and susceptibility
exhibit power-law when approaching the critical point and the corresponding
critical exponents respectably obey the Rushbrooke
inequality (RI) . Their analogues in percolation,
however, remain elusive. We define entropy, specific heat and redefine
susceptibility for percolation and show that they behave exactly in the same
way as their thermal counterpart. We also show that RI holds for both the
lattices albeit they belong to different universality classes.Comment: 5 pages, 3 captioned figures, to appear as a Rapid Communication in
Physical Review E, 201
Overfrustrated and Underfrustrated Spin-Glasses in d=3 and 2: Evolution of Phase Diagrams and Chaos Including Spin-Glass Order in d=2
In spin-glass systems, frustration can be adjusted continuously and
considerably, without changing the antiferromagnetic bond probability p, by
using locally correlated quenched randomness, as we demonstrate here on
hypercubic lattices and hierarchical lattices. Such overfrustrated and
underfrustrated Ising systems on hierarchical lattices in d=3 and 2 are
studied. With the removal of just 51 % of frustration, a spin-glass phase
occurs in d=2. With the addition of just 33 % frustration, the spin-glass phase
disappears in d=3. Sequences of 18 different phase diagrams for different
levels of frustration are calculated in both dimensions. In general,
frustration lowers the spin-glass ordering temperature. At low temperatures,
increased frustration favors the spin-glass phase (before it disappears) over
the ferromagnetic phase and symmetrically the antiferromagnetic phase. When any
amount, including infinitesimal, frustration is introduced, the chaotic
rescaling of local interactions occurs in the spin-glass phase. Chaos increases
with increasing frustration, as seen from the increased positive value of the
calculated Lyapunov exponent , starting from when
frustration is absent. The calculated runaway exponent of the
renormalization-group flows decreases with increasing frustration to
when the spin-glass phase disappears. From our calculations of entropy and
specific heat curves in d=3, it is seen that frustration lowers in temperature
the onset of both long- and short-range order in spin-glass phases, but is more
effective on the former. From calculations of the entropy as a function of
antiferromagnetic bond concentration p, it is seen that the ground-state and
low-temperature entropy already mostly sets in within the ferromagnetic and
antiferromagnetic phases, before the spin-glass phase is reached.Comment: Published version, 18 phase diagrams, 12 figures, 10 page
A domain-theoretic investigation of posets of sub-sigma-algebras (extended abstract)
Given a measurable space (X, M) there is a (Galois) connection between
sub-sigma-algebras of M and equivalence relations on X. On the other hand
equivalence relations on X are closely related to congruences on stochastic
relations. In recent work, Doberkat has examined lattice properties of posets
of congruences on a stochastic relation and motivated a domain-theoretic
investigation of these ordered sets. Here we show that the posets of
sub-sigma-algebras of a measurable space do not enjoy desired domain-theoretic
properties and that our counterexamples can be applied to the set of smooth
equivalence relations on an analytic space, thus giving a rather unsatisfactory
answer to Doberkat's question
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