To send content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about sending content to .
To send content items to your Kindle, first ensure email@example.com
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about sending to your Kindle.
Note you can select to send to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
An abelian processor is an automaton whose output is independent of the order of its inputs. Bond and Levine have proved that a network of abelian processors performs the same computation regardless of processing order (subject only to a halting condition). We prove that any finite abelian processor can be emulated by a network of certain very simple abelian processors, which we call gates. The most fundamental gate is a toppler, which absorbs input particles until their number exceeds some given threshold, at which point it topples, emitting one particle and returning to its initial state. With the exception of an adder gate, which simply combines two streams of particles, each of our gates has only one input wire, which sends letters (‘particles’) from a unary alphabet. Our results can be reformulated in terms of the functions computed by processors, and one consequence is that any increasing function from ℕk to ℕℓ that is the sum of a linear function and a periodic function can be expressed in terms of (possibly nested) sums of floors of quotients by integers.
We prove that proper coloring distinguishes between block factors and finitely dependent stationary processes. A stochastic process is finitely dependent if variables at sufficiently well-separated locations are independent; it is a block factor if it can be expressed as an equivariant finite-range function of independent variables. The problem of finding non-block-factor finitely dependent processes dates back to 1965. The first published example appeared in 1993, and we provide arguably the first natural examples. Schramm proved in 2008 that no stationary 1-dependent 3-coloring of the integers exists, and asked whether a
-coloring exists for any
. We give a complete answer by constructing a 1-dependent 4-coloring and a 2-dependent 3-coloring. Our construction is canonical and natural, yet very different from all previous schemes. In its pure form it yields precisely the two finitely dependent colorings mentioned above, and no others. The processes provide unexpected connections between extremal cases of the Lovász local lemma and descent and peak sets of random permutations. Neither coloring can be expressed as a block factor, nor as a function of a finite-state Markov chain; indeed, no stationary finitely dependent coloring can be so expressed. We deduce extensions involving
dimensions and shifts of finite type; in fact, any nondegenerate shift of finite type also distinguishes between block factors and finitely dependent processes.
Two related issues are explored for bond percolation on
(with d ≥ 3) and its dual plaquette process. Firstly, for what values of the parameter p does the complement of the infinite open cluster possess an infinite component? The corresponding critical point pfin satisfies pfin ≥ pc, and strict inequality is proved when either d is sufficiently large, or d ≥ 7 and the model is sufficiently spread out. It is not known whether d ≥ 3 suffices. Secondly, for what p does there exist an infinite dual surface of plaquettes? The associated critical point psurf satisfies psurf ≥ pfin.
be a discrete set in
. Call the elements of
centers. The well-known Voronoi tessellation partitions
into polyhedral regions (of varying volumes) by allocating each site of
to the closest center. Here we study allocations of
in which each center attempts to claim a region of equal volume
We focus on the case where
arises from a Poisson process of unit intensity. In an earlier paper by the authors it was proved that there is a unique allocation which is stable in the sense of the Gale–Shapley marriage problem. We study the distance
from a typical site to its allocated center in the stable allocation.
The model exhibits a phase transition in the appetite
. In the critical case
we prove a power law upper bound on
. (Power law lower bounds were proved earlier for all
). In the non-critical cases
we prove exponential upper bounds on
We consider the bond percolation model on the three-dimensional cubic lattice, in which individual edges are retained independently with probability p. We shall describe a subgraph of the lattice as ‘entangled’ if, roughly speaking, it cannot be ‘pulled apart’ in three dimensions. We shall discuss possible ways of turning this into a rigorous definition of entanglement. For a broad class of such definitions, we shall prove that for p sufficiently close to zero, the graph of retained edges has no infinite entangled subgraph almost surely, thereby establishing that there is a phase transition for entanglement at some value of p strictly between zero and unity.
We study finite and infinite entangled graphs
in the bond percolation process in three dimensions
with density $p$.
After a discussion of the relevant definitions,
the entanglement critical probabilities are defined.
The size of the maximal entangled graph at the origin
is studied for small $p$, and it is shown that this
graph has radius whose tail decays at least as fast
as $\exp(-\alpha n/\log n)$; indeed, the logarithm
may be replaced by any iterate of logarithm
for an appropriate positive constant $\alpha$. We
explore the question of almost sure uniqueness of
the infinite maximal open entangled graph when $p$
is large, and we establish two relevant theorems.
We make several conjectures concerning the properties
of entangled graphs in percolation. http://www.statslab.cam.ac.uk/$\sim$grg/
1991 Mathematics Subject Classification: primary 60K35;
secondary 05C10, 57M25, 82B41, 82B43, 82D60.
Email your librarian or administrator to recommend adding this to your organisation's collection.