22 research outputs found
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
Four-manifolds, geometries and knots
The goal of this book is to characterize algebraically the closed 4-manifolds
that fibre nontrivially or admit geometries in the sense of Thurston, or which
are obtained by surgery on 2-knots, and to provide a reference for the topology
of such manifolds and knots. The first chapter is purely algebraic. The rest of
the book may be divided into three parts: general results on homotopy and
surgery (Chapters 2-6), geometries and geometric decompositions (Chapters
7-13), and 2-knots (Chapters 14-18). In many cases the Euler characteristic,
fundamental group and Stiefel-Whitney classes together form a complete system
of invariants for the homotopy type of such manifolds, and the possible values
of the invariants can be described explicitly. The strongest results are
characterizations of manifolds which fibre homotopically over S^1 or an
aspherical surface (up to homotopy equivalence) and infrasolvmanifolds (up to
homeomorphism). As a consequence 2-knots whose groups are poly-Z are determined
up to Gluck reconstruction and change of orientations by their groups alone.
This book arose out of two earlier books "2-Knots and their Groups" and "The
Algebraic Characterization of Geometric 4-Manifolds", published by Cambridge
University Press for the Australian Mathematical Society and for the London
Mathematical Society, respectively. About a quarter of the present text has
been taken from these books, and I thank Cambridge University Press for their
permission to use this material. The book has been revised in March 2007. For
details see the end of the preface.Comment: This is the revised version published by Geometry & Topology
Monographs in March 200
Random Embeddings of Graphs: The Expected Number of Faces in Most Graphs is Logarithmic
A random 2-cell embedding of a connected graph in some orientable surface
is obtained by choosing a random local rotation around each vertex. Under this
setup, the number of faces or the genus of the corresponding 2-cell embedding
becomes a random variable. Random embeddings of two particular graph classes --
those of a bouquet of loops and those of parallel edges connecting two
vertices -- have been extensively studied and are well-understood. However,
little is known about more general graphs despite their important connections
with central problems in mainstream mathematics and in theoretical physics (see
[Lando & Zvonkin, Springer 2004]). There are also tight connections with
problems in computing (random generation, approximation algorithms). The
results of this paper, in particular, explain why Monte Carlo methods (see,
e.g., [Gross & Tucker, Ann. NY Acad. Sci 1979] and [Gross & Rieper, JGT 1991])
cannot work for approximating the minimum genus of graphs.
In his breakthrough work ([Stahl, JCTB 1991] and a series of other papers),
Stahl developed the foundation of "random topological graph theory". Most of
his results have been unsurpassed until today. In our work, we analyze the
expected number of faces of random embeddings (equivalently, the average genus)
of a graph . It was very recently shown [Campion Loth & Mohar, arXiv 2022]
that for any graph , the expected number of faces is at most linear. We show
that the actual expected number of faces is usually much smaller. In
particular, we prove the following results:
1) , for
sufficiently large. This greatly improves Stahl's upper bound for
this case.
2) For random models containing only graphs, whose maximum
degree is at most , we show that the expected number of faces is
.Comment: 44 pages, 6 figure
The combinatorics of the Jack parameter and the genus series for topological maps
Informally, a rooted map is a topologically pointed embedding of a graph in a surface. This thesis examines two problems in the enumerative theory of rooted maps.
The b-Conjecture, due to Goulden and Jackson, predicts that structural similarities between the generating series for rooted orientable maps with respect
to vertex-degree sequence, face-degree sequence, and number of edges, and
the corresponding generating series for rooted locally orientable maps, can be
explained by a unified enumerative theory. Both series specialize M(x,y,z;b), a
series defined algebraically in terms of Jack symmetric functions, and the unified
theory should be based on the existence of an appropriate integer valued invariant of rooted maps with respect to which M(x,y,z;b) is the generating series for locally orientable maps. The conjectured invariant should take the value zero when evaluated on orientable maps, and should take positive values when evaluated on non-orientable maps, but since it must also depend on
rooting, it cannot be directly related to genus.
A new family of candidate invariants, η, is described recursively in terms of root-edge deletion. Both the generating series for rooted maps with respect to η and an appropriate specialization of M satisfy the same differential equation with a unique solution. This shows that η gives the appropriate enumerative theory when vertex degrees are ignored, which is precisely the setting required by Goulden, Harer, and Jackson for an application to algebraic geometry. A functional equation satisfied by M and the existence of a bijection between
rooted maps on the torus and a restricted set of rooted maps on the Klein bottle show that η has additional structural properties that are required of the conjectured invariant.
The q-Conjecture, due to Jackson and Visentin, posits a natural combinatorial
explanation, for a functional relationship between a generating series for rooted
orientable maps and the corresponding generating series for 4-regular rooted
orientable maps. The explanation should take the form of a bijection, ϕ, between appropriately decorated rooted orientable maps and 4-regular rooted orientable
maps, and its restriction to undecorated maps is expected to be related to the
medial construction.
Previous attempts to identify ϕ have suffered from the fact that the existing
derivations of the functional relationship involve inherently non-combinatorial
steps, but the techniques used to analyze η suggest the possibility of a new derivation of the relationship that may be more suitable to combinatorial analysis. An examination of automorphisms that must be induced by ϕ gives evidence for a refinement of the functional relationship, and this leads to a more combinatorially refined conjecture. The refined conjecture is then reformulated algebraically so that its predictions can be tested numerically
Combinatorial and Algebraic Enumeration: a survey of the work of Ian P. Goulden and David M. Jackson
Progress in Surface Theory
The theory of surfaces is interpreted these days as a prototype of submanifold geometry and is characterized by the substantial application of PDE methods and methods from the theory of integrable systems, in addition to the more classical techniques from real and/or complex analysis. In addition, surfaces with singularities are studied intensively. In this workshop we brought together all the main strands of modern surface theory
Thick hyperbolic 3-manifolds with bounded rank
We construct a geometric decomposition for the convex core of a thick
hyperbolic 3-manifold M with bounded rank. Corollaries include upper bounds in
terms of rank and injectivity radius on the Heegaard genus of M and on the
radius of any embedded ball in the convex core of M.Comment: 170 pages, 17 figure