24,471 research outputs found
Number of cycles in the graph of 312-avoiding permutations
The graph of overlapping permutations is defined in a way analogous to the De
Bruijn graph on strings of symbols. That is, for every permutation there is a directed edge from the
standardization of to the standardization of
. We give a formula for the number of cycles of
length in the subgraph of overlapping 312-avoiding permutations. Using this
we also give a refinement of the enumeration of 312-avoiding affine
permutations and point out some open problems on this graph, which so far has
been little studied.Comment: To appear in the Journal of Combinatorial Theory - Series
Cardinality of Rauzy classes
Rauzy classes define a partition of the set of irreducible (or
indecomposable) permutations. They were defined by G. Rauzy as part of an
induction algorithm for interval exchange transformations. In this article we
prove an explicit formula for the cardinality of all Rauzy classes.Comment: 43 pages, 22 figure
Universal behaviour of 3D loop soup models
These notes describe several loop soup models and their {\it universal
behaviour} in dimensions greater or equal to 3. These loop models represent
certain classical or quantum statistical mechanical systems. These systems
undergo phase transitions that are characterised by changes in the structures
of the loops. Namely, long-range order is equivalent to the occurrence of
macroscopic loops. There are many such loops, and the joint distribution of
their lengths is always given by a {\it Poisson-Dirichlet distribution}.
This distribution concerns random partitions and it is not widely known in
statistical physics. We introduce it explicitly, and we explain that it is the
invariant measure of a mean-field split-merge process. It is relevant to
spatial models because the macroscopic loops are so intertwined that they
behave effectively in mean-field fashion. This heuristics can be made exact and
it allows to calculate the parameter of the Poisson-Dirichlet distribution. We
discuss consequences about symmetry breaking in certain quantum spin systems.Comment: 31 pages, 11 figures. Notes prepared for the 6th Warsaw School of
Statistical Physics, held from 25 June to 2 July 2016 in Sandomierz, Polan
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