80,254 research outputs found
Property (T) and rigidity for actions on Banach spaces
We study property (T) and the fixed point property for actions on and
other Banach spaces. We show that property (T) holds when is replaced by
(and even a subspace/quotient of ), and that in fact it is
independent of . We show that the fixed point property for
follows from property (T) when 1
. For simple Lie groups and their lattices, we prove that the fixed point property for holds for any if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive Banach spaces.Comment: Many minor improvement
Lower matching conjecture, and a new proof of Schrijver's and Gurvits's theorems
Friedland's Lower Matching Conjecture asserts that if is a --regular
bipartite graph on vertices, and denotes the number of
matchings of size , then where . When
, this conjecture reduces to a theorem of Schrijver which says that a
--regular bipartite graph on vertices has at least
perfect matchings. L. Gurvits
proved an asymptotic version of the Lower Matching Conjecture, namely he proved
that
In this paper, we prove the Lower Matching Conjecture. In fact, we will prove
a slightly stronger statement which gives an extra factor
compared to the conjecture if is separated away from and , and is
tight up to a constant factor if is separated away from . We will also
give a new proof of Gurvits's and Schrijver's theorems, and we extend these
theorems to --biregular bipartite graphs
The rigidity of infinite graphs
A rigidity theory is developed for the Euclidean and non-Euclidean placements
of countably infinite simple graphs in R^d with respect to the classical l^p
norms, for d>1 and 1<p<\infty. Generalisations are obtained for the Laman and
Henneberg combinatorial characterisations of generic infinitesimal rigidity for
finite graphs in the Euclidean plane. Also Tay's multi-graph characterisation
of the rigidity of generic finite body-bar frameworks in d-dimensional
Euclidean space is generalised to the non-Euclidean l^p norms and to countably
infinite graphs. For all dimensions and norms it is shown that a generically
rigid countable simple graph is the direct limit of an inclusion tower of
finite graphs for which the inclusions satisfy a relative rigidity property.
For d>2 a countable graph which is rigid for generic placements in R^d may fail
the stronger property of sequential rigidity, while for d=2 the equivalence
with sequential rigidity is obtained from the generalised Laman
characterisations. Applications are given to the flexibility of non-Euclidean
convex polyhedra and to the infinitesimal and continuous rigidity of compact
infinitely-faceted simplicial polytopes.Comment: 51 page
Density classification on infinite lattices and trees
Consider an infinite graph with nodes initially labeled by independent
Bernoulli random variables of parameter p. We address the density
classification problem, that is, we want to design a (probabilistic or
deterministic) cellular automaton or a finite-range interacting particle system
that evolves on this graph and decides whether p is smaller or larger than 1/2.
Precisely, the trajectories should converge to the uniform configuration with
only 0's if p1/2. We present solutions to that problem
on the d-dimensional lattice, for any d>1, and on the regular infinite trees.
For Z, we propose some candidates that we back up with numerical simulations
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