477 research outputs found
The nematic phase of a system of long hard rods
We consider a two-dimensional lattice model for liquid crystals consisting of
long rods interacting via purely hard core interactions, with two allowed
orientations defined by the underlying lattice. We rigorously prove the
existence of a nematic phase, i.e., we show that at intermediate densities the
system exhibits orientational order, either horizontal or vertical, but no
positional order. The proof is based on a two-scales cluster expansion: we
first coarse grain the system on a scale comparable with the rods' length; then
we express the resulting effective theory as a contour's model, which can be
treated by Pirogov-Sinai methods.Comment: 36 pages, 4 figures; abstract changed, references added, comparison
with literature improved, figures adde
Transience of Edge-Reinforced Random Walk
We show transience of the edge-reinforced random walk (ERRW) for small
reinforcement in dimension d greater than 2. This proves the existence of a
phase transition between recurrent and transient behavior, thus solving an open
problem stated by Diaconis in 1986. The argument adapts the proof of
quasi-diffusive behavior of the SuSy hyperbolic model for fixed conductances by
Disertori, Spencer and Zirnbauer [CMP 2010], using the representation of ERRW
as a mixture of vertex-reinforced jump processes (VRJP) with independent gamma
conductances, and the interpretation of the limit law of VRJP as a
supersymmetric (SuSy) hyperbolic sigma model developed by Sabot and Tarr\`es in
[JEMS 2014].Comment: 25 pages, 2 figure
Plate-nematic phase in three dimensions
We consider a system of anisotropic plates in the three-dimensional
continuum, interacting via purely hard core interactions. We assume that the
particles have a finite number of allowed orientations. In a suitable range of
densities, we prove the existence of a uni-axial nematic phase, characterized
by long range orientational order (the minor axes are aligned parallel to each
other, while the major axes are not) and no translational order. The proof is
based on a coarse graining procedure, which allows us to map the plate model
into a contour model, and in a rigorous control of the resulting contour
theory, via Pirogov-Sinai methods.Comment: 29 pages, 4 figure
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