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Shift-Coupling of Random Rooted Graphs and Networks
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Abstract
In this paper, we present a result similar to the shift-coupling result of Thorisson (1996) in the context of random graphs and networks. The result is that a given random rooted network can be obtained by changing the root of another given one if and only if the distributions of the two agree on the invariant sigma-field. Several applications of the result are presented for the case of unimodular networks. In particular, it is shown that the distribution of a unimodular network is uniquely determined by its restriction to the invariant sigma-filed. Also, the theorem is applied to the existence of an invariant transport kernel that balances between two given (discrete) measures on the vertices. An application is the existence of a so called extra head scheme for the Bernoulli process on an infinite unimodular graph. Moreover, a construction is presented for balancing transport kernels that is a generalization of the Gale-Shapley stable matching algorithm in bipartite graphs. Another application is on a general method that covers the situations where some vertices and edges are added to a unimodular network and then, to make it unimodular, the probability measure is biased and then a new root is selected. It is proved that this method provides all possible unimodularizations in these situations. Finally, analogous existing results for stationary point processes and unimodular networks are discussed in detail.
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We give an algorithm to construct a translation-invariant transport kernel
between ergodic stationary random measures and on ,
given that they have equal intensities. Our algorithm is deterministic given
realizations and of the measures. The existence of such a
transport kernel was proved by Thorisson and Last. Our algorithm is a
generalization of the work of Hoffman, Holroyd and Peres, which settles the
case that is the Lebesgue measure and is a point process. To do
so, we limit ourselves to mild transport kernels which are obtained from a
measure whose density w.r.t. is bounded by 1. We give a
definition of stability of mild transport kernels and introduce a construction
algorithm inspired by the Gale-Shapley stable marriage algorithm. For stable
mild transport kernels, we study existence, uniqueness, monotonicity w.r.t. the
measures and boundedness. As a result, our algorithm yields a construction of a
shift-coupling of an ergodic random measure and its Palm version.
. Let G be a closed group of automorphisms of a graph X. We relate geometric properties of G and X, such as amenability and unimodularity, to properties of G-invariant percolation processes on X, such as the number of innite components, the expected degree, and the topology of the components. Our fundamental tool is a new mass-transport technique that has been occasionally used elsewhere and is developed further here. Perhaps surprisingly, these investigations of group-invariant percolation produce results that are new in the Bernoulli setting. Most notably, we prove that critical Bernoulli percolation on any nonamenable Cayley graph has no innite clusters. More generally, the same is true for any nonamenable graph with a unimodular transitive automorphism group. We show that G is amenable i for all < 1, there is a G-invariant site percolation process ! on X with P[x 2 !] > for all vertices x and with no innite components. When G is not amenable, a threshold < 1 ...
Let be a discrete set in . Call the elements of centers. The well-known Voronoi tessellation partitions into polyhedral regions (of varying sizes) by allocating each site of to the closest center. Here we study ``fair'' allocations of to in which the regions allocated to different centers have equal volumes. We prove that if is obtained from a translation-invariant point process, then there is a unique fair allocation which is stable in the sense of the Gale--Shapley marriage problem. (I.e., sites and centers both prefer to be allocated as close as possible, and an allocation is said to be unstable if some site and center both prefer each other over their current allocations.) We show that the region allocated to each center is a union of finitely many bounded connected sets. However, in the case of a Poisson process, an infinite volume of sites are allocated to centers further away than . We prove power law lower bounds on the allocation distance of a typical site. It is an open problem to prove any upper bound in .
Let \Pi be an ergodic simple point process on R^d and let \Pi^* be its Palm version. Thorisson [Ann. Probab. 24 (1996) 2057-2064] proved that there exists a shift coupling of \Pi and \Pi^*; that is, one can select a (random) point Y of \Pi such that translating \Pi by -Y yields a configuration whose law is that of \Pi^*. We construct shift couplings in which Y and \Pi^* are functions of \Pi, and prove that there is no shift coupling in which \Pi is a function of \Pi^*. The key ingredient is a deterministic translation-invariant rule to allocate sets of equal volume (forming a partition of R^d) to the points of \Pi. The construction is based on the Gale-Shapley stable marriage algorithm [Amer. Math. Monthly 69 (1962) 9-15]. Next, let \Gamma be an ergodic random element of {0,1}^{Z^d} and let \Gamma^* be \Gamma conditioned on \Gamma(0)=1. A shift coupling X of \Gamma and \Gamma^* is called an extra head scheme. We show that there exists an extra head scheme which is a function of \Gamma if and only if the marginal E[\Gamma(0)] is the reciprocal of an integer. When the law of \Gamma is product measure and d\geq3, we prove that there exists an extra head scheme X satisfying E\exp c\|X\|^d<\infty; this answers a question of Holroyd and Liggett [Ann. Probab. 29 (2001) 1405-1425].
A factor graph of a point process is a graph whose vertices are the points of the process, and which is constructed from the process in a deterministic isometry-invariant way. We prove that the d-dimensional Poisson process has a one-ended tree as a factor graph. This implies that the Poisson points can be given an ordering isomorphic to the usual ordering of the integers in a deterministic isometry-invariant way. For d \geq 4 our result answers a question posed by Ferrari, Landim and Thorisson. We prove also that any isometry-invariant ergodic point process of finite intensity in Euclidean or hyperbolic space has a perfect matching as a factor graph provided all the inter-point distances are distinct.
A basic dichotomy concerning the structure of the orbit space of a transformation group has been discovered by Glimm [G12] in the locally compact group action case and extended by Effros [E 1, E2] in the Polish group action
case when additionally the induced equivalence relation is Fσ. It is the purpose of this paper to extend the Glimm-Effros dichotomy to the very general context of an arbitrary Borel equivalence relation (not even necessarily induced by a group action). Despite the totally classical descriptive set-theoretic nature of our result, our proof requires the employment of methods of effective descriptive
set theory and thus ultimately makes crucial use of computability (or recursion) theory on the integers.
Let (X, B) be a standard Borel space, R Ì X x X an equivalence. relation £& x Assume each equivalence class is countable. Theorem 1:3 a countable group G of Borel isomorphisms of (X, $) so that R - {(*, gx): g Î G. G is far from unique. However, notions like invariance and quasi-invariance and R-N derivatives of measures depend only on R, not the choice of G. We develop some of the ideas of Dye [1], [2] and Krieger [1]—[5] in a fashion explicitly avoiding any choice of G; we also show the connection with virtual groups. A notion of “module over R” is defined, and we axiomatize and develop a cohomology theory for R with coefficients in such a module. Surprising application (contained in Theorem 7): let a, p be rationally independent irrationals on the circle T, and /Borel: T->T. Then 3 Borel g, h: T-*T with f(x) - (g(ax)/g(x))(h(bx)/h(x)) a.e. The notion of “skew product action*' is generalized to our context, and provides a setting for a generalization of the Krieger invariant for the R-N derivative of an ergodic transformation: we define, for a cocycle c on R with values in the group A, a subgroup of A depending only on the cohomology class of c, and in Theorem 8 identify this with another subgroup, the “normalized proper range” of c, defined in terms of the skew action. See also Schmidt [1].
Foundations of Stochastic Geometry.- Prolog.- Random Closed Sets.- Point Processes.- Geometric Models.- Integral Geometry.- Averaging with Invariant Measures.- Extended Concepts of Integral Geometry.- Integral Geometric Transformations.- Selected Topics from Stochastic Geometry.- Some Geometric Probability Problems.- Mean Values for Random Sets.- Random Mosaics.- Non-stationary Models.- Facts from General Topology.- Invariant Measures.- Facts from Convex Geometry.
Introduction This survey describes a general approach to a class of problems that arise in combinatorial probability and combinatorial optimization. Formally, the method is part of weak convergence theory, but in concrete problems the method has a flavor of its own. A characteristic element of the method is that it often calls for one to introduce a new, infinite, probabilistic object whose local properties inform us about the limiting properties of a sequence of finite problems. The name objective method hopes to underscore the value of shifting ones attention to the new, large random object with fixed distributional properties and way from the sequence of objects with changing distributions. The new object always provides us with some new information on the asymptotic behavior of the original sequence, and, in the happiest cases, the constants associated with the infinite object even permit us to find the elusive limit constants for that sequence. 1.1 A Motivating Example: the Ass
We introduce and study invariant (weighted) transport-kernels balancing stationary random measures on a locally compact Abelian group. The first main result is an associated fundamental invariance property of Palm measures, derived from a generalization of Neveu's exchange formula. The second main result is a simple sufficient and necessary criterion for the existence of balancing invariant transport-kernels. We then introduce (in a nonstationary setting) the concept of mass-stationarity with respect to a random measure, formalizing the intuitive idea that the origin is a typical location in the mass. The third main result of the paper is that a measure is a Palm measure if and only if it is mass-stationary. Comment: Published in at http://dx.doi.org/10.1214/08-AOP420 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
Consider a locally compact second countable topological transformation group acting measurably on an arbitrary space. We show that the distributions of two random elements X and in this space agree on invariant sets if and only if there is a random transformation such that has the same distribution as . Applying this to random fields in d dimensions under site shifts, we show further that these equivalent claims are also equivalent to site-average total variation convergence. This convergence result extends to amenable groups.
Thorisson and others have proved results that imply the following: given an i.i.d. family of Bernoulli random variables indexed byZ
d
there exists an occupied siteX ∈Z
d
with the property that relative to its location, the other variables are still i.i.d. We raise the question of how large such anXmust be. Ford =1, we prove that anyXwith this property satisfies E | X |½= ∞. Moreover, there does exist such anXwith tails P(|X| 1 ≥ n) of orderCn
-½so these results are essentially best possible. Analogous results for the Poisson process in one dimension are given. The corresponding problem for stationary ergodic sequences is considered also. This project was motivated by some tagged particle problems.
We show that when percolation produces infinitely many infinite clusters on a Cayley graph, one cannot distinguish the clusters from each other by any invariantly defined property. This implies that uniqueness of the infinite cluster is equivalent to nondecay of connectivity (a.k.a. long-range order). We then derive applications concerning uniqueness in Kazhdan groups and in wreath products and inequalities for p
u
.
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.
- F Baccelli
- M O Haji-Mirsadeghi
- A Khezeli
Baccelli, F., Haji-Mirsadeghi, M. O., and Khezeli, A. (2016). Eternal Family Trees and Dynamics on Unimodular Random Graphs. arXiv preprint arXiv:1608.05940.
Mass transport between stationary random measures (Doctoral dissertation
- A Khezeli
Khezeli, A. (2016). Mass transport between stationary random measures (Doctoral dissertation), Sharif University of Technology.
Stationäre zufällige Maße auf lokalkompakten abelschen Gruppen. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete
- J Mecke
Mecke, J. (1967). Stationäre zufällige Maße auf lokalkompakten abelschen Gruppen. Zeitschrift
für Wahrscheinlichkeitstheorie und verwandte Gebiete, 9(1), 36-58.