13,738 research outputs found
Regularity lemmas in a Banach space setting
Szemer\'edi's regularity lemma is a fundamental tool in extremal graph
theory, theoretical computer science and combinatorial number theory. Lov\'asz
and Szegedy [L. Lov\'asz and B. Szegedy: Szemer\'edi's Lemma for the analyst,
Geometric and Functional Analysis 17 (2007), 252-270] gave a Hilbert space
interpretation of the lemma and an interpretation in terms of compact- ness of
the space of graph limits. In this paper we prove several compactness results
in a Banach space setting, generalising results of Lov\'asz and Szegedy as well
as a result of Borgs, Chayes, Cohn and Zhao [C. Borgs, J.T. Chayes, H. Cohn and
Y. Zhao: An Lp theory of sparse graph convergence I: limits, sparse random
graph models, and power law distributions, arXiv preprint arXiv:1401.2906
(2014)].Comment: 15 pages. The topological part has been substantially improved based
on referees comments. To appear in European Journal of Combinatoric
No directed fractal percolation in zero area
We show that fractal (or "Mandelbrot") percolation in two dimensions produces
a set containing no directed paths, when the set produced has zero area. This
improves a similar result by the first author in the case of constant retention
probabilities to the case of retention probabilities approaching 1
Cardy's Formula for Certain Models of the Bond-Triangular Type
We introduce and study a family of 2D percolation systems which are based on
the bond percolation model of the triangular lattice. The system under study
has local correlations, however, bonds separated by a few lattice spacings act
independently of one another. By avoiding explicit use of microscopic paths, it
is first established that the model possesses the typical attributes which are
indicative of critical behavior in 2D percolation problems. Subsequently, the
so called Cardy-Carleson functions are demonstrated to satisfy, in the
continuum limit, Cardy's formula for crossing probabilities. This extends the
results of S. Smirnov to a non-trivial class of critical 2D percolation
systems.Comment: 49 pages, 7 figure
Partition function zeros at first-order phase transitions: Pirogov-Sinai theory
This paper is a continuation of our previous analysis [BBCKK] of partition
functions zeros in models with first-order phase transitions and periodic
boundary conditions. Here it is shown that the assumptions under which the
results of [BBCKK] were established are satisfied by a large class of lattice
models. These models are characterized by two basic properties: The existence
of only a finite number of ground states and the availability of an appropriate
contour representation. This setting includes, for instance, the Ising, Potts
and Blume-Capel models at low temperatures. The combined results of [BBCKK] and
the present paper provide complete control of the zeros of the partition
function with periodic boundary conditions for all models in the above class.Comment: 46 pages, 2 figs; continuation of math-ph/0304007 and
math-ph/0004003, to appear in J. Statist. Phys. (special issue dedicated to
Elliott Lieb
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