7 research outputs found
On Topological Minors in Random Simplicial Complexes
For random graphs, the containment problem considers the probability that a
binomial random graph contains a given graph as a substructure. When
asking for the graph as a topological minor, i.e., for a copy of a subdivision
of the given graph, it is well-known that the (sharp) threshold is at .
We consider a natural analogue of this question for higher-dimensional random
complexes , first studied by Cohen, Costa, Farber and Kappeler for
.
Improving previous results, we show that is the
(coarse) threshold for containing a subdivision of any fixed complete
-complex. For higher dimensions , we get that is an
upper bound for the threshold probability of containing a subdivision of a
fixed -dimensional complex.Comment: 15 page
Dichromatic polynomials and Potts models summed over rooted maps
We consider the sum of dichromatic polynomials over non-separable rooted
planar maps, an interesting special case of which is the enumeration of such
maps. We present some known results and derive new ones. The general problem is
equivalent to the -state Potts model randomized over such maps. Like the
regular ferromagnetic lattice models, it has a first-order transition when
is greater than a critical value , but is much larger - about 72
instead of 4.Comment: 29 pages, three figures changes in App D, introduction and
acknowledgement
Enumeration of simple bipartite maps on the sphere and the projective plane
AbstractThis paper is concerned with the number of rooted simple bipartite maps on the plane according to the root-face valencies and the number of edges of the maps. An explicit formula with one parameter is given. Furthermore, a special kind of rooted simple bipartite maps on the projective plane are counted and, as special cases, recursive formulae for maps with fewer faces on the projective plane are presented as well
Asymptotic Expansions for Sub-Critical Lagrangean Forms
Asymptotic expansions for the Taylor coefficients of the Lagrangean form phi(z)=zf(phi(z)) are examined with a focus on the calculations of the asymptotic coefficients. The expansions are simple and useful, and we discuss their use in some enumerating sequences in trees, lattice paths and planar maps
Probability around the Quantum Gravity. Part 1: Pure Planar Gravity
In this paper we study stochastic dynamics which leaves quantum gravity
equilibrium distribution invariant. We start theoretical study of this dynamics
(earlier it was only used for Monte-Carlo simulation). Main new results concern
the existence and properties of local correlation functions in the
thermodynamic limit. The study of dynamics constitutes a third part of the
series of papers where more general class of processes were studied (but it is
self-contained), those processes have some universal significance in
probability and they cover most concrete processes, also they have many
examples in computer science and biology. At the same time the paper can serve
an introduction to quantum gravity for a probabilist: we give a rigorous
exposition of quantum gravity in the planar pure gravity case. Mostly we use
combinatorial techniques, instead of more popular in physics random matrix
models, the central point is the famous exponent.Comment: 40 pages, 11 figure