Christopher Hoffman

A dyadic endomorphism whose square is Bernoulli (2008)

ω-Periodic graphs are introduced and studied. These are graphs which arise as the limits of periodic extensions of the nearest neighbor graph on the integers. We observe that all bounded degree...

Uniform Endomorphisms Which Are Isomorphic To A Bernoulli Shift (2007)

Abstract. One of the corollaries of Ornstein’s isomorphism theorem is that if (Y, S, ν) is an invertible measure preserving transformation and (Y, S 2, ν) is isomorphic to a Bernoulli shift then...

Uniform endomorphisms which are isomorphic to a Bernoulli shift (2007)

. A uniformly p to one endomorphism is a measure preserving map with entropy log p which is a.e. p to 1 and for which the conditional expectation of each preimage is precisely 1=p. The standard...

Uniform endomorphisms which are isomorphic to a Bernoulli shift (2007)

Abstract. A uniformly p to one endomorphism is a measure preserving map with entropy log p which is a.e. p to 1 and for which the conditional expectation of each preim-age is precisely 1/p. The...

A ZERO ENTROPY T SUCH THAT THE [T,ID] ENDOMORPHISM IS NONSTANDARD (2007)

Abstract. A uniformly p to one endomorphism is a measure preserving map with entropy log p which is a.e. p to 1 and for which the conditional expectation of each preimage is precisely 1=p. The...

Percolation Clusters, Christopher Hoffman

Abstract. We present an example of an ergodic transformation T, a variant of a zero entropy non loosely Bernoulli map of Feldman [1], such that the sequence of random variables generated by the...

d (2007)

Grimmett, Kesten, and Zhang (1993) showed that for d 3 and p? p c (Z

A Family of Nonisomorphic (2007)

Markov Random Fields, Christopher Hoffman

It was recently shown that there exists a family Z Markov random fields which are K but are not isomorphic to Bernoulli shifts [4]. In this paper we show that most distinct members of this family are...

Rational Maps are d-adic Bernoulli (2007)

Phase Transition in Dependent Percolation (2007)

Freire, Lopes and Mane proved that for any rational map f there exists a natural invariant measure f [5]. Mane showed there exists an n > 0 such that (f , f ) is measurably conjugate to the...

If the [T, Id] automorphism is Bernoulli then (2007)

In this paper we discuss two di#erent models of dependent percolation on the graph . We show that they both exhibit phase transitions. This proves a conjecture of Jonasson, Mossel and Peres [6], who...

The Id Endomorphism, Christopher Hoffman, Daniel Rudolph

For any 1-1 measure preserving map T of a probability space we can form the [T , Id] and [T , T -1 ] automorphisms as well as the corresponding endomorphisms and decreasing sequence of #-algebras. In...

Energy of flows on Z (2007)

Percolation Clusters Christopher, Christopher Hoffman

We show that if p > p c (Z ), then the unique infinite percolation cluster supports a nonzero flow f with finite q energy for all q > 2. This extends the work of Grimmett, Kesten, and Zhang and...

A Zero Entropy T Such That The [t ,id] (2007)

Endomorphism Is Nonstandard, Christopher Hoffman

We present an example of an ergodic transformation T , a variant of a zero entropy non loosely Bernoulli map of Feldman [1], such that the sequence of random variables generated by the [T ,Id]...

Entropy and Dyadic Equivalence of Random (2007)

Walks On Random, Deborah Heicklen, Christopher Hoffman, Daniel Rudolph

For any 1-1 measure preserving map T of a probability space, consider the [T , T -1 ] endomorphism and the corresponding decreasing sequence of #-algebras. We demonstrate that if the decreasing...

A dyadic endomorphism which is Bernoulli (2007)

But Not Standard, Christopher Hoffman, Daniel Rudolph

Any measure preserving endomorphism generates both a decreasing sequence of #-algebras and an invertible extension. In this paper we exhibit a dyadic measure preserving endomorphism (X, T , ) such...

The fundamental group of random 2-complexes (2007)

Babson, Eric, Hoffman, Christopher, Kahle, Matthew

The random 2-complex Y=Y(n,p) is the probability space of all simplicial complexes on vertex set [n] and edge set [n] \choose 2, with each 2-dimensional face included with probability p...

Exponential clogging time for a one dimensional DLA (2007)

Benjamini, Itai, Hoffman, Christopher

When considering DLA on a cylinder it is natural to ask how many particles it takes to clog the cylinder, e.g. modeling clogging of arteries. In this note we formulate a very simple DLA clogging...

A special set of exceptional times for dynamical random walk on $\Z^2$ (2006)

Amir, Gideon, Hoffman, Christopher

Benjamini,Haggstrom, Peres and Steif introduced the model of dynamical random walk on Z^d. This is a continuum of random walks indexed by a parameter t. They proved that for d=3,4 there almost surely...

A stable marriage of Poisson and Lebesgue (2006)

Hoffman, Christopher, Holroyd, Alexander E., Peres, Yuval

Let Ξ be a discrete set in ℝd. Call the elements of Ξ centers. The well-known Voronoi tessellation partitions ℝd into polyhedral regions (of varying sizes) by allocating each site of ℝd to...

Geodesics in First Passage Percolation (2005)

Tail Bounds for the Stable Marriage of Poisson and Lebesgue (2005)

We consider a wide class of ergodic first passage percolation processes on Z^2 and prove that there exist at least four one-sided geodesics a.s. We also show that coexistence is possible with...

Hoffman, Christopher, Holroyd, Alexander E., Peres, Yuval

Let \Xi be a discrete set in R^d. Call the elements of \Xi centers. The well-known Voronoi tessellation partitions R^d into polyhedral regions (of varying volumes) by allocating each site of R^d to...

A stable marriage of Poisson and Lebesgue (2005)

Hoffman, Christopher, Holroyd, Alexander E., Peres, Yuval

Let $\Xi$ be a discrete set in ${\mathbb{R}}^d$. Call the elements of $\Xi$ centers. The well-known Voronoi tessellation partitions ${\mathbb{R}}^d$ into polyhedral regions (of varying sizes) by...

Mixing times of the biased card shuffling and the asymmetric exclusion process (2005)

Benjamini, Itai, Berger, Noam, Hoffman, Christopher, Mossel, Elchanan

Consider the following method of card shuffling. Start with a deck of N cards numbered 1 through N. Fix a parameter p between 0 and 1. In this model a "shuffle" consists of uniformly selecting a pair...

Return Probabilities of a Simple Random Walk on Percolation Clusters (2005)

Heicklen, Deborah; Lockhead-Martin, USA; Deborah.w.heicklen@lmco.com, Hoffman, Christopher; University Of Washington, USA; Hoffman@math.washington.edu

We bound the probability that a continuous time simple random walk on the infinite percolation cluster on Zd returns to the origin at time t. We use this result to show that in dimensions 5 and...

Recurrence of Simple Random Walk on $Z^2$ is Dynamically Sensitive (2005)

Coexistence for Richardson type competing spatial growth models (2005)

Benjamini, Haggstrom, Peres and Steif introduced the concept of a dynamical random walk. This is a continuous family of random walks, {S_n(t)}. Benjamini et. al. proved that if d=3 or d=4 then there...

Mixing times of the biased card shuffling and the asymmetric exclusion process (2005)

We study a large family of competing spatial growth models. In these models the vertices in ℤd can take on three possible states {0,1,2}. Vertices in states 1 and 2 remain in their states forever,...

Itai Benjamini, Noam Berger, Christopher Hoffman

Abstract. Consider the following method of card shuffling. Start with a deck of N cards numbered 1 through N. Fix a parameter p between 0 and 1. In this model a “shuffle ” consists of uniformly...

Return probabilities of a simple random walk on percolation clusters (2005)

Return probabilities of a simple random walk on percolation clusters (2005)

We show that the probability that a continuous time simple random walk on the infinite percolation cluster in Z d for d ≥ 2 returns to the origin at time t is less than or equal to Ct −d/2 (log...

Uniform endomorphisms which are isomorphic to a Bernoulli shift (2004)

Benjamini and Schramm [6] showed that the probability that a simple random walk on the infinite percolation cluster in Z d returns to the origin at time t is greater than or equal to Ct \Gammad=2. In...

Hoffman, Christopher, Rudolph, Daniel

A {\it uniformly $p$-to-one endomorphism} is a measure-preserving map with entropy log $p$ which is almost everywhere $p$-to-one and for which the conditional expectation of each preimage is...

Coexistence for Richardson type competing spatial growth models (2004)

Nonuniqueness for specifications in $\ell^{2+\epsilon}$ (2003)

We study a large family of competing spatial growth models. In these the vertices in Z^d can take on three possible states {0,1,2}. Vertices in states 1 and 2 remain in their states forever, while...

Berger, Noam, Hoffman, Christopher, Sidoravicius, Vladas

For every $p>2$, we construct a regular and continuous specification ($g$-function), which has a variation sequence that is in $l^p$ and which admits multiple Gibbs measures. Combined with a recent...

The scenery factor of the [T, T −1 ] transformation is not loosely Bernoulli (2003)

Mixing times of the biased card shuffling and the asymmetric exclusion process (2002)

Kalikow proved that the [T , T -1 ] transformation is not isomorphic to a Bernoulli shift [3]. We show that the scenery factor of the [T , T -1 ] transformation is not isomorphic to a Bernoulli...

Benjamini, Itai, Berger, Noam, Hoffman, Christopher, Mossel, Elchanan

Consider the following method of card shuffling. Start with a deck of $N$ cards numbered 1 through N. Fix a parameter $p$ between 0 and 1. In this model a ``shuffle'' consists of uniformly selecting...

Uniform endomorphisms which are isomorphic to a Bernoulli shift (2002)

Hoffman, Christopher, Rudolph, Daniel

A {\it uniformly $p$\/{\rm -}\/to\/{\rm -}\/one endomorphism} is a measure-preserving map with entropy log $p$ which is almost everywhere $p$-to-one and for which the conditional expectation of each...

Rational maps are d-adic Bernoulli (2002)

Heicklen, Deborah, Hoffman, Christopher

Freire, Lopes and Mañé proved that for any rational map $f$ there exists a natural invariant measure $\mu_f$. Mañé showed there exists an $n>0$ such that $(f^n, \mu_f)$ is measurably conjugate to...

Mixing Times of the biased card shuffling and the asymmetric exclusion process (2002)

Itai Benjamini, Noam Berger, Christopher Hoffman, Elchanan Mossel

Consider the following method of card shu#ing. Start with a deck of N cards numbered 1 through N . Fix a parameter p between 0 and 1. In this model a "shu#e" consists of uniformly selecting...

A Markov random field which is K but not Bernoulli (1999)

A K counterxample machine (1999)

We use a variant of a discrete time exclusion process, studied by Yaguchi [7], to construct a Markov random field which is K but not Bernoulli. Instead of having all the particles in the exclusion...

A dyadic endomorphism which is Bernoulli but not standard (1999)

Abstract. We present a general method for constructing families of measure preserving transformations which are K and loosely Bernoulli with various ergodic theoretical properties. For example, we...

If the [T,Id] auotmorphism is Bernoulli then the [T,Id] endomorphism is Standard (1999)

Any measure preserving endomorphism generates both a decreasing sequence of oe-algebras and an invertible (two sided) extension. In this paper we describe a dyadic measure preserving endomorphism (X;...

A K counterxample machine (1999)

For any 1-1 measure preserving map T of a probability space we can form the [T ; Id] automorphism as well as the [T ; Id] and [T ; T \Gamma1 ] endomorphisms and the corresponding decreasing sequence...

Unpredictable nearest neighbor processes (1998)

Abstract. We present a general method for constructing families of measure preserving transformations which are K and loosely Bernoulli with various ergodic theoretical proper-ties. For example, we...

Energy of flows on percolation clusters (1998)

Benjamini, Pemantle and Peres constructed nearest neighbor processes which have predictability profiles that decay faster than that of the simple random walk. Häggström and Mossel found processes...

Unpredictable nearest neighbor processes (1998)

It is well known for which gauge functions H there exists a flow on Z d with finite H energy. In this paper we discuss the robustness under random thinning of edges of the existence of such flows....

T, T^-1 is not standard (1998)

Abstract. Benjamini, Pemantle, and Peres constructed nearest neighbor processes which have predictability profiles that decay faster than that of the simple random walk. Haggstrom and Mossel found...

Energy of flows on percolation clusters (1998)

A sequence of random variables, Y 0; Y 1; Y 2;:::, is called standard if there exists a one sided isomorphism between it and a sequence of independent random variables. In this paper it is...

Energy of flows on percolation clusters (1998)

It is well known for which gauge functions H there exists a flow on Z d with finite H energy. In this paper we discuss the robustness under random thinning of edges of the existence of such flows....

An uncountable family of nonisomorphic Markov random fields which are K but not Bernoulli (1998)

It is well known for which gauge functions H there exists a flow on Z d with finite H energy. In this paper we discuss the robustness under random thinning of edges of the existence of such flows....

Entropy and Dyadic Equivalence of Random Walks on a Random Scenery (1998)

It was recently shown that there exists a family Z 2 Markov random fields which are K but are not isomorphic to Bernoulli shifts [6]. In this paper we show that most distinct members of this family...

Deborah Heicklen, Christopher Hoffman, Daniel Rudolph

For any 1-1 measure preserving map T of a probability space consider the [T ; T \Gamma1 ] endomorphism and the corresponding decreasing sequence of oe-algebras. We demonstrate that if the decreasing...

Rational Maps are 1-Sided Bernoulli (1998)

Energy of Flows on Percolation Clusters Christopher Homan (1998)

In [5], Freire, Lopes and Ma~n'e proved that for any rational map f there exists a natural invariant measure ¯ f . In 1985, Ma~n'e showed there exists an n ? 0 such that (f n ; ¯ f ) is...

T, T^-1 is not standard (1998)

It is well known for which gauge functions H there exists a flow on Z with finite H energy. In this paper we discuss the robustness under random thinning of edges of the existence of such flows....

A loosely Bernoulli counterexample machine (1997)

A sequence of random variables, Y0, Y1, Y2,..., is called standard if there exists a one sided isomorphism between it and a sequence of independent random variables. In this paper it is demonstrated...

The Behavior of Bernoulli Shifts Relative to their Factors (1997)

In Rudolph's paper on minimal self joinings [7] he proves that a rank one mixing transformation constructed by Ornstein [5] can be used as the building block for many ergodic theoretical...

A loosely Bernoulli counterexample machine (1997)

We present numerous examples of ways that a Bernoulli shift can behave relative to a family of factors. This shows the similarities between the properties which collections of ergodic transformations...