367 research outputs found
Processes on Unimodular Random Networks
We investigate unimodular random networks. Our motivations include their
characterization via reversibility of an associated random walk and their
similarities to unimodular quasi-transitive graphs. We extend various theorems
concerning random walks, percolation, spanning forests, and amenability from
the known context of unimodular quasi-transitive graphs to the more general
context of unimodular random networks. We give properties of a trace associated
to unimodular random networks with applications to stochastic comparison of
continuous-time random walk.Comment: 66 pages; 3rd version corrects formula (4.4) -- the published version
is incorrect --, as well as a minor error in the proof of Proposition 4.10;
4th version corrects proof of Proposition 7.1; 5th version corrects proof of
Theorem 5.1; 6th version makes a few more minor correction
Doeblin Trees
This paper is centered on the random graph generated by a Doeblin-type
coupling of discrete time processes on a countable state space whereby when two
paths meet, they merge. This random graph is studied through a novel subgraph,
called a bridge graph, generated by paths started in a fixed state at any time.
The bridge graph is made into a unimodular network by marking it and selecting
a root in a specified fashion. The unimodularity of this network is leveraged
to discern global properties of the larger Doeblin graph. Bi-recurrence, i.e.,
recurrence both forwards and backwards in time, is introduced and shown to be a
key property in uniquely distinguishing paths in the Doeblin graph, and also a
decisive property for Markov chains indexed by . Properties related
to simulating the bridge graph are also studied.Comment: 44 pages, 4 figure
The Superparticle and the Lorentz Group
We present a unified group-theoretical framework for superparticle theories.
This explains the origin of the ``twistor-like'' variables that have been used
in trading the superparticle's -symmetry for worldline supersymmetry.
We show that these twistor-like variables naturally parametrise the coset space
, where is the Lorentz group
and is its maximal subgroup. This space is a compact manifold, the
sphere . Our group-theoretical construction gives the proper
covariantisation of a fixed light-cone frame and clarifies the relation between
target-space and worldline supersymmetries.Comment: 33 page
Noncommutative Geometry as a Framework for Unification of all Fundamental Interactions including Gravity. Part I
We examine the hypothesis that space-time is a product of a continuous
four-dimensional manifold times a finite space. A new tensorial notation is
developed to present the various constructs of noncommutative geometry. In
particular, this notation is used to determine the spectral data of the
standard model. The particle spectrum with all of its symmetries is derived,
almost uniquely, under the assumption of irreducibility and of dimension 6
modulo 8 for the finite space. The reduction from the natural symmetry group
SU(2)xSU(2)xSU(4) to U(1)xSU(2)xSU(3) is a consequence of the hypothesis that
the two layers of space-time are finite distance apart but is non-dynamical.
The square of the Dirac operator, and all geometrical invariants that appear in
the calculation of the heat kernel expansion are evaluated. We re-derive the
leading order terms in the spectral action. The geometrical action yields
unification of all fundamental interactions including gravity at very high
energies. We make the following predictions: (i) The number of fermions per
family is 16. (ii) The symmetry group is U(1)xSU(2)xSU(3). (iii) There are
quarks and leptons in the correct representations. (iv) There is a doublet
Higgs that breaks the electroweak symmetry to U(1). (v) Top quark mass of
170-175 Gev. (v) There is a right-handed neutrino with a see-saw mechanism.
Moreover, the zeroth order spectral action obtained with a cut-off function is
consistent with experimental data up to few percent. We discuss a number of
open issues. We prepare the ground for computing higher order corrections since
the predicted mass of the Higgs field is quite sensitive to the higher order
corrections. We speculate on the nature of the noncommutative space at
Planckian energies and the possible role of the fundamental group for the
problem of generations.Comment: 56 page
Unimodular Random Trees
We consider unimodular random rooted trees (URTs) and invariant forests in
Cayley graphs. We show that URTs of bounded degree are the same as the law of
the component of the root in an invariant percolation on a regular tree. We use
this to give a new proof that URTs are sofic, a result of Elek. We show that
ends of invariant forests in the hyperbolic plane converge to ideal boundary
points. We also prove that uniform integrability of the degree distribution of
a family of finite graphs implies tightness of that family for local
convergence, also known as random weak convergence.Comment: 19 pages, 4 figure
- …