Recent papers

B. G. Franzen, J. Steif and J. Wästlund
Where to stand when playing darts?,
ALEA. Latin American Journal of Probability and Mathematical Statistics,
18, (2021), no. 2, 15611583.
Abstract.
This paper analyzes the question of where one should stand when playing
darts.
If one stands at distance d>0 and aims at a in R^n,
then the dart (modelled by a random
vector X in R^n)
hits a random point given by a+dX. Next, given a payoff function
f, one considers
\sup_a Ef(a+dX)
and asks if this is decreasing in d; i.e.,
whether it is
better to stand closer rather than farther
from the target. Perhaps surprisingly, this is
not always the case and understanding
when this does or does not occur is the purpose
of this paper.
We show that if X has a socalled selfdecomposable distribution, then
it is always better
to stand closer for any payoff function. This class includes all stable
distributions as
well as many more.
On the other hand, if the payoff function is cos(x),
then it is always better to stand closer
if and only if the characteristic function phi_X(t)
is decreasing on [0,\infty).
We will then
show that if there are at least two point masses,
then it is not always better to stand closer
using cos(x).
If there is a single point mass, one can find a different payoff function to
obtain this phenomenon.
Another large class of darts X for which there are bounded continuous payoff functions for
which
it is not always better to stand closer are distributions with compact
support. This
will be obtained by using the fact that the Fourier transform of such distributions
has a zero
in the complex plane.
This argument will work whenever there is a complex zero of the
Fourier transform.
Finally, we analyze if the property of it being better to stand closer is
closed under
convolution and/or limits.
 M. Palö Forsström and J. Steif
Divide and color representations for threshold Gaussian and stable vectors,
Electronic Journal of Probability, Vol. 25, (2020) paper no. 54, 145.
Abstract.
We study the question of when a {0,1}valued threshold process associated
to a mean zero Gaussian or a symmetric stable vector corresponds to a divide and
color
(DC) process. This means that the process corresponding to fixing a threshold level h
and
letting a 1 correspond to the variable being larger than h arises from a random
partition
of the index set followed by coloring {\it all} elements in each
partition element 1 or 0
with probabilities p and 1p, independently for different partition elements.
While it turns out that all discrete Gaussian free fields yield a DC process when the
threshold
is zero, for general $n$dimensional mean zero, variance one
Gaussian vectors with nonnegative
covariances, this is true in general when $n=3$ but is false for n=4.
The behavior is quite different depending on whether the threshold level h is zero or
not and we
show that there is no general monotonicity in h in either direction.
We also show that all constant
variance discrete Gaussian free fields with a
finite number of variables yield DC processes for large
thresholds.
In the stable case, for the simplest nontrivial symmetric stable vector with three
variables, we obtain
a phase transition in the stability exponent alpha at the
surprising value of 1/2; if the index of stability
is larger than $1/2$, then the process
yields a DC process for large $h$ while if the index of stability
is smaller than
1/2, then this is not the case.

M. Palö Forsström and J. Steif
An analysis of the induced linear operators associated to divide and color
models,
Journal of Theoretical Probability 34, (2021), 10431060.
Abstract.
We study the natural linear operators associated to divide and color (DC) models.
The degree
of nonuniqueness of the random partition yielding a DC model is directly
related to the dimension of
the kernel of these linear operators. We determine exactly the dimension of these
kernels as well as
analyze
a permutationinvariant version. We also obtain properties of the solution set for
certain parameter
values which will be important in (1) showing that large
threshold discrete Gaussian free fields are DC
models and in (2) analyzing
when the Ising model with a positive external field is a DC model, both in
future work.
However, even here, we give an application to the Ising model on a triangle.

M. Palö Forsström and J. Steif
A formula for hidden regular variation behavior for symmetric
stable distributions,
Extremes 23 (2020), no. 4, 667691.
Abstract.
We develop a formula for the powerlaw decay of various sets for
symmetric stable random
vectors
in terms of how many vectors from the support of the corresponding spectral measure are
needed to enter the set.
One sees different decay rates in "different directions", illustrating the
phenomenon of "hidden regular variation". We give several examples and obtain quite varied
behavior,
including sets which do not have exact powerlaw decay.

M. Palö Forsström and J. Steif
A few surprising integrals,
Statistics and Probability Letters, 157, (2020), 108635.
Abstract.
Using formulas for certain quantities involving stable vectors, due to
I. Molchanov, and in
some cases utilizing the socalled divide and color model,
we prove that certain families of integrals
which, ostensibly, depend on a parameter are in fact independent of this
parameter.

M. Palö Forsström and J. Steif
FortuinKasteleyn representations for threshold Gaussian and stable vectors
,
(This paper has been replaced by the four papers above. However, there is still
some material in this
paper which is not contained in any of the above four
papers.)
Abstract.
We study the question of when a 0,1valued threshold process associated
to a mean zero
Gaussian or a symmetric stable vector corresponds to a
"divide and color (DC) process".
This means
that the process corresponding to fixing a threshold level h and letting a 1 correspond
to the variable
being larger than h arises from a random partition of the index set followed by
coloring all elements
in each partition element 1 or 0 with probabilities p and 1p,
independently for different partition
elements. For example, the Ising model with zero external
field (as well as a number of other well
known processes) is a DC process where
the random partition corresponds to the famous FKrandom
cluster model.
We first determine in general the exact lack of uniqueness of the representing random partition.
This
amounts to determining the dimensions of the kernels of a certain family of linear operators.
We obtain various results in both the Gaussian and symmetric stable cases. For example, it turns
out that
all discrete Gaussian free fields yield a DC process when the threshold is
zero; this follows quite easily
from known facts. For general ndimensional mean zero, variance one
Gaussian vectors with nonnegative
covariances, the zerothreshold process is always a DC
process for n=3 but this is false for n=4.
The answers in general are quite different depending on whether the threshold level h is zero or not.
We show
that there is no general monotonicity in h in either direction. We also show that all discrete
Gaussian free fields
yield DC processes for large thresholds.
Moving to n=3, we characterize exactly which mean zero, variance one
Gaussian vectors yield
DC processes for large h.
In the stable case, among other results, if we stick to the simplest case of a permutation
invariant, symmetric
stable vector with three variables, we obtain
a phase transition in the stability exponent \alpha at the surprising
point 1/2; if the index
of stability is larger than 1/2, then the process yields
a DC process for large h while if
the index of stability is smaller than 1/2,
then this is not the case.

Y. Peres, P. Sousi and J. Steif
Mixing time for random walk on supercritical dynamical percolation
,
Probability Theory and Related Fields, 176, (2020), 809849.
Abstract.
We consider dynamical percolation on the ddimensional discrete torus of side
length
n, Z_n^d, where each edge refreshes its status at rate
\mu=\mu_n\le 1/2 to be open with
probability p. We study random walk on
the torus, where the walker moves at rate 1/(2d)
along each open edge.
In earlier work of two of the authors with A. Stauffer, it was shown that
in the subcritical case p < p_c(Z^d),
the (annealed) mixing time of the walk is Theta(n^2/mu),
and it was conjectured that in the supercritical case p p_c(Z^d),
the mixing time is Theta(n^2+1/mu);
here the implied constants depend only on d and p.
We prove a quenched (and hence annealed) version of
this conjecture up to a
polylogarithmic factor under the assumption theta(p)>1/2. Our proof
is based on percolation results (e.g., the GrimmettMarstrand Theorem) and an
analysis of the
volumebiased evolving set process; the key point is that typically,
the evolving set has a substantial
intersection with the giant percolation cluster
at many times. This allows us to use precise
isoperimetric properties of the cluster
(due to G. Pete) to infer rapid growth of the evolving set,
which in turn yields
the upper bound on the mixing time.

Y. Peres, P. Sousi and J. Steif
Quenched exit times for random walk on dynamical percolation
,
Markov Processes and Related Fields, 24, (2018), 715731.
Abstract.
We consider random walk on dynamical percolation on the discrete torus Z_n^d.
In previous
work, mixing times of this process for p < p_c(Z^d)
were obtained in the annealed
setting where one
averages over the dynamical percolation environment. Here
we study exit times in the quenched setting,
where we condition on a typical dynamical
percolation environment. We obtain an upper bound for all p
which for p < p_c
matches the known lower bound.

J. Steif and J. Tykesson
Generalized Divide and Color models ,
ALEA. Latin American Journal of Probability and Mathematical Statistics,
16, (2019), 899955.
Abstract.
In this paper, we initiate the study of ``Generalized Divide and Color Models''.
A very
special interesting case of this is the
``Divide and Color Model'' (which motivates the name we
use)b
introduced and studied by Olle Häggström.
In this generalized model, one starts with a finite or countable set
V, a random partition of V
and a parameter p in [0,1]. The corresponding
generalized Divide and Color Model is the
{0,1}valued process indexed by V obtained by
independently, for each partition element in
the random partition
chosen, with probability p, assigning all the elements of the partition element
the value 1,
and with probability 1p assigning all the elements of the partition element the value 0.
Some of the questions which we study here are the following. Under what situations can different
random partitions give rise to the same color process?
What can one say concerning exchangeable
random partitions?
What is the set of product measures that a color process stochastically dominates?
For random partitions which are translation invariant, what ergodic properties
do the resulting color
processes have?
The motivation for studying these processes is twofold; on the one hand, we believe that this is
a very
natural and interesting class of processes that deserves investigation and
on the other hand, a number
of quite varied wellstudied processes
actually fall into this class such as (1) the Ising model,
(2) the
fuzzy Potts model, (3) the stationary distributions for the Voter Model,
(4) random walk in random
scenery and of course (5) the original Divide and Color Model.

J. Jonasson and J. Steif
Volatility of Boolean functions ,
Stochastic Processes and Their Applications,
Volume 126, (2016), 29562975.
Abstract.
We study the volatility of the output of a Boolean function when the input
bits undergo
a natural dynamics. For n = 1,2,...., let
f_n:{0,1}^{m_n} map to {0,1} be a Boolean function and
X^{(n)}(t)=(X_1(t),\ldots,X_{m_n}(t))_{t \in [0,infty)} be a vector of i.i.d.
stationary continuous
time Markov chains on {0,1} that jump from 0
to 1 with rate
p_n \in [0,1] and from 1 to 0 with rate
q_n=1p_n. Our object of study will be
C_n which is the number of state changes of f_n(X^{(n)}(t))
as a function of
t during [0,1].
We say that the family {f_n}_{n\ge 1} is volatile if C_n tends to infinity
in distribution as n tends to infty and say that
{f_n}_{n\ge 1} is tame if {C_n}_{n\ge 1} is tight.
We
study these concepts in and of themselves as well as investigate their
relationship
with the recent
notions of noise sensitivity and noise stability. In addition, we
study the question of lameness which
means that \Pro(C_n =0) tends to 1
as n\to\infty.
Finally, we investigate these properties for the majority
function, iterated 3majority, the AND/OR function on the binary tree
and percolation on certain trees at
various levels of the parameter p_n.

T. Cox, Y. Peres and J. Steif
Cutoff for the noisy voter model
,
Annals of Applied Probability, Volume 26, (2016), 917932.
Abstract. Given a continuous time Markov Chain {q (x, y )} on a finite
set S , the associated noisy
voter model is the continuous time Markov
chain on {0, 1}^S which evolves by (1) for each two sites
x and y in S ,
the state at site x changes to the value of the state at site y at rate
q (x, y ) and (2) each
site rerandomizes its state at rate 1. We show that
if there is a uniform bound on the rates {q (x, y )}
and the corresponding
stationary distributions are "almost" uniform, then the mixing time has
a
sharp cutoff at time log S /2 with a window of order 1. Lubetzky and
Sly proved cutoff with a
window of order 1 for the stochastic Ising model
on toroids: we obtain the special case of their result
for the cycle as a
consequence of our result. Finally, we consider the model on a star and
demonstrate
the surprising phenomenon that the time it takes for the
chain started at all ones to become close in total
variation to the chain
started at all zeros is of smaller order than the mixing time.

D. Ahlberg and J. Steif (with an appendix by Gabor Pete)
Scaling limits for the threshold window:
When does a monotone Boolean function flip its outcome?
,
Annales Institut Henri Poincare, Probabilites et Statistiques,
53, (2017), 21352161.
Consider a monotone Boolean function f:{0,1}^n \to {0,1} and the canonical
monotone coupling
{eta_p:p in [0,1]} of an element in {0,1}^n chosen according to product measure with intensity
p in [0,1]. The random point p in [0,1] where f(eta_p) flips from 0 to 1
is often concentrated
near a particular point, thus exhibiting a threshold phenomenon. For a sequence of such Boolean functions,
we peer closely into this threshold window and consider, for large n, the limiting distribution
(properly
normalized to be nondegenerate) of this random point where the Boolean function
switches from being 0 to 1.
We determine this distribution for a number of the Boolean functions which are
typically studied and pay
particular attention to the functions corresponding
to iterated majorityand percolation crossings.
It turns out
that these limiting distributions have quite varying behavior. In fact, we show
that any nondegenerate
probability measure on R arises in this way for some sequence of Boolean functions.

A. E. Holroyd, Y. Peres and J. Steif
Wald for nonstopping times: The rewards of impatient prophets,
Electronic Communications in Probability, 19, (2014), no. 78, 9pp.
Let $X_1,X_2,\ldots$ be independent identically distributed nonnegative
random variables. Wald's identity
states that the random sum
$S_T:=X_1+\cdots+X_T$ has expectation $\E T \cdot \E X_1$ provided $T$
is a
stopping time. We prove here that for any $1<\alpha\leq 2$,
if $T$ is an arbitrary nonnegative random
variable, then
$S_T$ has finite expectation provided that
$X_1$ has finite $\alpha$moment and $T$ has
finite $1/(\alpha1)$moment.
We also prove a variant in which $T$ is assumed to have a finite exponential
moment. These moment conditions are sharp in the sense that for any i.i.d.\ sequence
$X_i$ violating them,
there is a $T$ satisfying the given condition for which
$S_T$ (and, in fact, $X_T$) has infinite expectation.
An interpretation of this is given in terms of a prophet being more rewarded than a gambler
when a certain
impatience restriction is imposed.

Y. Peres, A. Stauffer and J. Steif
Random walks on dynamical percolation: mixing times, mean
squared displacement and hitting times,
Probability Theory and Related Fields, 162, (2015), 487530.
We study the behavior of random walk on dynamical percolation. In this model,
the edges of a graph G
are either open or closed and
refresh their status at rate mu while at the same time a random walker moves
on G at rate 1
but only along edges which are open.
On the ddimensional torus with side length n, we
prove that in the subcritical regime,
the mixing times for both the full system and the random walker are
n^2/mu up to constants. We also obtain results concerning mean squared
displacement and hitting times.
Finally, we show that the usual recurrence transience dichotomy
for the lattice Z^d holds for this model as well.

E. Lubetzky and J. Steif
Strong noise sensitivity and random graphs,
Annals of Probability, 43, (2015), 32393278.
The noise sensitivity of a Boolean function describes its
likelihood to flip under small
perturbations of its input. Introduced in
the seminal work of Benjamini, Kalai and
Schramm (1999), it was there
shown to be governed by the first level of Fourier
coefficients in the
central case of monotone functions at a constant critical probability.
Here we study noise sensitivity and a natural stronger version of it,
addressing the effect
of noise given a specific witness in the
original input.
Our main context is the ErdosRenyi
random graph, where already the
property of containing a given graph is sufficiently rich
to separate these
notions. In particular, our analysis implies (strong) noise sensitivity in
settings where the BKS criterion involving the first Fourier level does not
apply, e.g., when
the critical value goes to 0 polynomially fast in the number of variables.

S. Blachere, F. den Hollander and J. Steif
A crossover for the bad configurations of random walk in random scenery,
Annals of Probability, 39, (2011), 20182041.
In this paper we consider a random walk and a random color scenery on Z. The
increments
of the walk and the colors of the scenery are assumed to be i.i.d.
and to be independent of each
other. We are interested in the random process of
colors seen by the walk in the course of time.
Bad configurations for this random
process are the discontinuity points of the conditional probability
distribution
for the color seen at time zero given the colors seen at all later times.
We focus on the case where the random walk has increments 0, +1 or 1 with
probability
epsilon, (1epsilon)p and (1epsilon)(1p), respectively, with p in
[1/2,1] and epsilon in [0,1),
and where the scenery assigns the color black or
white to the sites of Z with probability 1/2
each. We show that, remarkably,
the set of bad configurations exhibits a crossover: for epsilon=0 and
p in (1/2,4/5),
all configurations are bad, while for (p,epsilon) in an open neighborhood
of (1,0) all configurations are good. In addition, we show that for epsilon=0 and
p=1/2 both
bad and good configurations exist. We conjecture that for all \epsilon
in [0,1) the crossover value is
unique and equals 4/5. Finally, we
suggest an approach to handle the seemingly more difficult
epsilon>0 and
p in (1/2,4/5) case, which will be pursued in future work.

C. Garban and J. Steif
Noise sensitivity and percolation,
Probability and statistical physics in two and more dimensions,
49154, Clay Math. Proc., 15,
Amer. Math. Soc., Providence, RI, 2012.
The present text provides the lecture notes for the course
"noise sensitivity and percolation"
held for the 2010 Clay summer school.

J. Steif
A mini course on percolation theory,
Jyväskylä Lectures in Mathematics, 3, 2011.
These are lecture notes based on a mini course on percolation which was
given at the Jyväskylä
summer school in mathematics in
Jyväskylä, Finland, August 2009. The point of the course was
to try to touch on a number of different topics in
percolation in order to give people some feel for
the field. These
notes follow fairly closely the lectures given in the summer school.
However, some
topics covered in these notes were not covered in the lectures
(such as continuity of the percolation
function above the critical
value) while other topics covered in detail in the lectures
are not proved
in these notes (such as conformal invariance).

E. Broman, C. Garban and J. Steif
Exclusion Sensitivity of Boolean Functions,
Probability Theory and Related Fields, 155, (2013), 621663.
Recently the study of noise sensitivity and noise stability of Boolean
functions has received
considerable attention. The purpose of this paper is to
extend these notions in
a natural way
to a different class of perturbations, namely those arising
from running the symmetric
exclusion process for a short amount of time.
In this study, the case of monotone
Boolean
functions will turn out to be of particular interest. We show that
for this class of functions,
ordinary noise sensitivity and noise
sensitivity with respect to the complete graph exclusion
process are equivalent.
We also show this equivalence with respect to stability.
After obtaining these fairly general results, we study
``exclusion sensitivity'' of critical
percolation
in more detail with respect to
mediumrange dynamics.
The exclusion dynamics,
due to its conservative nature, is more physical than the
classical i.i.d. dynamics. Interestingly,
we will see
that in order to obtain a precise understanding of the exclusion sensitivity
of percolation,
we will need to describe how typical spectral sets of percolation
diffuse under
the underlying exclusion process.

J. Steif
A survey on dynamical percolation ,
Fractal geometry and stochastics, IV, Birkhauser, 2009, 145174.
Percolation is one of the simplest and nicest models in probability
theory/statistical
mechanics which exhibits critical phenomena.
Dynamical percolation is a model where
a simple
time dynamics is added to the (ordinary) percolation model.
This dynamical
model exhibits very interesting behavior.
Our goal in this
survey is to give an overview
of the work in dynamical
percolation that has been done (and some of
which is in the
process of being written up).

A. Bandyopadhyay, J. Steif and A. Timar
On the Cluster Size Distribution for Percolation on Some General Graphs,
Revista Matematica Iberoamericana, 26 (2010), 529550.
We show that for any Cayley graph, the probability (at any p)
that the cluster of the origin
has size n decays at a welldefined
exponential rate (possibly 0). For general graphs, we
relate this rate being positive in the supercritical regime with the
amenability/nonamenability
of the underlying graph.

J. Steif and M. Warfheimer
The critical contact process in a randomly evolving environment dies out,
ALEA. Latin American Journal of Probability and Mathematical Statistics,
4 (2008), 337357.
Bezuidenhout and Grimmett proved that the critical contact process dies out.
Here, we
generalize the result to the so called contact process in a
random evolving environment (CPREE),
introduced by Erik Broman. This process
is a generalization of the contact process where the
recovery rate can
vary between two values. The rate which it chooses is determined by a
background process, which evolves independently at different sites. As for
the contact process,
we can similarly define a critical value in terms of
survival for this process. In this paper we prove
that this definition is
independent of how we start the background process, that finite and
infinite
survival (meaning nontriviality of the upper invariant measure)
are equivalent and finally that the
process dies out at criticality.

Y. Peres, O. Schramm and J. Steif
Dynamical sensitivity of the infinite cluster in critical percolation,
Annales Institut Henri Poincare, Probabilites et Statistiques,
45 (2009), 491514
In dynamical percolation, the status of every bond is refreshed according to
an independent
Poisson clock.
For graphs which do not percolate at criticality, the dynamical sensitivity of
this property was analyzed
extensively in the last decade. Here we focus on graphs which
percolate at criticality,
and investigate the dynamical sensitivity of the infinite cluster.
We
first give two examples of bounded
degree graphs, one which percolates for all times at
criticality
and one which has exceptional times of
nonpercolation. We then make a nearly
complete analysis
of this question for spherically symmetric trees
with spherically symmetric
edge probabilities
bounded away from 0 and 1.
One interesting regime occurs when the
expected number of vertices at the
nth level that connect to the root at a fixed time is
of order
n(\log n)^\alpha.
R. Lyons (1990) showed that at a fixed time, there is an infinite cluster
a.s.
if and only if \alpha >1. We prove that the probability that there is
an infinite cluster at all times
is 1
if \alpha > 2, while this probability is 0 if 1<\alpha \le 2.
Within the regime where a.s. there
is an infinite cluster at all times,
there is yet another type of ``phase transition'' in the
behavior
of the process: if the expected number of
vertices at the nth level
connecting to the root at a fixed
time
is of order n^\theta with \theta > 2,
then the number of connected components of the set of
times
in [0,1] at which the root does not percolate
is finite a.s., while if 1<\theta < 2, then the
number of such
components is infinite with positive probability.

J. Jonasson and J. Steif
Dynamical models for circle covering: Brownian motion and Poisson updating,
Annals of Probability, 36, (2008), 739764.
(The published version is a slightly condensed version of what
appears here.)
We consider two dynamical variants of the classical problem of random
interval coverings
of the unit circle, the latter having been
completely solved by L. Shepp.
In the first model,
the centers of the intervals perform independent
Brownian motions and in the second model,
the positions of the
intervals are updated according to independent Poisson processes where
an interval
of length $\ell$ is updated at rate $\ell^{\alpha}$ where $\alpha$ is a parameter.
For the model with Brownian motions, a special case of our results is
that if the length of the
$n$th interval is $c/n$, then there are times
at which a fixed point is not covered if and only if
$c <2$ and
there are times at which the circle is not fully covered if and only if $c <3$.
For the
Poisson updating model, we obtain analogous results with
$c <\alpha$ and $c <\alpha +1$ instead.
We also compute the Hausdorff dimension of the set of exceptional times
for some of these questions.

J. Steif and A. Sudbury
Some results for poisoning in a catalytic model,
Electronic Communications in Probability,
11 (2006), 168177.
We obtain new results concerning poisoning/nonpoisoning in
a catalytic model which has
previously been introduced and studied.
We show that poisoning can occur even when the
arrival rate of one gas
is smaller than the sum of the arrival rates of the other gases,
and that
poisoning does not occur when all gases have equal
arrival rates.

T. Liggett, J. Steif and B. Toth
Statistical mechanical systems on complete graphs, infinite
exchangeability, finite extensions
and a discrete finite moment problem,
Annals of Probability, 35, (2007), 867914.
We show that a large collection of statistical mechanical systems with
quadratically
represented Hamiltonians
on the complete graph can be extended to infinite exchangeable
processes.
This includes all ferromagnetic Ising, Potts and Heisenberg models.
By
de Finetti's theorem, this is equivalent to showing that these
probability measures can be
expressed as averages of
product measures. We provide examples showing that
``ferromagnetism''
is not however in itself sufficient and also study in some detail the
Ising model
with an additional 3body interaction. Finally, we study the
question of how
much the antiferromagnetic Ising model can be extended.
In this direction, we obtain sharp
asymptotic results via a
solution to a new moment problem. We also obtain a ``formula'' for
the extension which is valid in many cases.

O. Schramm and J. Steif
Quantitative noise sensitivity and exceptional times for percolation,
Annals of Mathematics, 171, (2010), 619672.
One goal of this paper is to prove that dynamical critical site percolation on the
planar
triangular lattice
has exceptional times at which percolation occurs.
In doing so, new
quantitative noise sensitivity results
for percolation are obtained.
The latter is based on a
novel method for controlling the
``level $k$'' Fourier coefficients via the construction of
a
randomized
algorithm which looks at random bits, outputs the value of a
particular
function but looks at any fixed input bit with low
probability. We also obtain upper and
lower bounds on the Hausdorff
dimension of the set of percolating times. We then study
the problem
of exceptional times for certain
``$k$arm'' events on wedges and cones. As
a corollary
of this analysis, we prove, among other things, that there are no times
at which
there are two infinite ``white'' clusters, obtain an upper bound
on the Hausdorff dimension
of the set of times at which there are
both an infinite white cluster and an infinite black cluster
and prove that for dynamical critical bond percolation on the square grid
there are no exceptional
times at which $3$ disjoint infinite clusters are
present.

F. den Hollander and J. Steif
Random walk in random scenery: A survey of some recent results,
Dynamics and Stochastics:
Festschrift in Honor of Michael Keane,
IMS Lecture NotesMonograph
Series, Vol. 48 (2006) 5365.
In this paper we give a survey of some recent results for random walk in
random
scenery (RWRS). On $\Z^d$, $d\geq 1$, we are given a random walk
with i.i.d.
increments and a random scenery with i.i.d.\ components. The
walk and the scenery
are assumed to be independent. RWRS is the random
process where time is indexed
by $\Z$, and at each unit of time both the
step taken by the walk and the scenery value
at the site that is visited
are registered. We collect various results that classify the
ergodic
behavior of RWRS in terms of the characteristics of the underlying random
walk (and discuss extensions to stationary walk increments and stationary
scenery
components as well). We describe a number of results for scenery
reconstruction and
close by listing some open questions.

E. I. Broman, O. Häggström and J. Steif
Refinements of stochastic domination,
Probability Theory and Related Fields, 136 (2006), 587603.
In a recent paper by two of the authors,
the concepts of upwards and downwards
$\epsilon$movability were introduced, mainly as a
technical tool for studying
dynamical percolation of
interacting particle systems.
In this paper, we further
explore these concepts which can be seen as
refinements or quantifications of stochastic
domination, and we
relate them to previously studied concepts such as uniform
insertion
tolerance and extractability.

E. Mossel, R. O'Donnell, O. Regev, J. Steif and B. Sudakov
Noninteractive correlation distillation,
inhomogeneous Markov
chains, and the reverse
BonamiBeckner inequality,
Israel Journal of Mathematics, 154 (2006), 299336.
In this paper we study noninteractive
correlation distillation (NICD), a generalization of
noise
sensitivity previously studied earlier.
We extend the model to NICD on trees. In this
model there
is a fixed undirected tree with players at some of the nodes. One
node is given
a uniformly random string and this string is
distributed throughout the network, with the
edges of the tree
acting as independent binary symmetric channels. The goal of the
players
is to agree on a shared random bit without communicating.
Our new contributions include the following:
(1). In the case of a $k$leaf star graph (the model considered
earlier by Mossel and O'Donnell),
we resolve the open question of whether the
success probability must go to zero as $k \to \infty$.
We show
that this is indeed the case and provide matching upper and lower
bounds on the
asymptotically optimal rate (a slowlydecaying
polynomial).
(2). In the case of the $k$vertex path graph, we show
that it is always optimal for all players to
use the same
1bit function.
(3). In the general case we show that all players should use
monotone functions. We also show,
somewhat surprisingly, that
for certain trees it is better if not all players use the same function.
Our techniques include the use of the reverse BonamiBeckner
inequality. Although the usual
BonamiBeckner has been frequently used
before,
its reverse counterpart seems very littleknown;
To demonstrate its strength, we use
it to prove a new isoperimetric inequality for the discrete
cube
and a new result on the mixing of short random walks on the cube.
Another tool that we need
is a tight bound on the
probability that a Markov chain stays inside certain sets; we
prove a new
theorem generalizing and strengthening previous such
bounds.
On the probabilistic side, we use
the ``reflection principle'' and the FKG
and related inequalities in order to study the problem on
general trees.

T. Liggett and J. Steif
Stochastic Domination: The Contact Process, Ising Models and FKG Measures
Annales Institut Henri Poincare, Probabilites et Statistiques,
42 (2006), 223243.
We prove for the contact process on $Z^d$, and many other
graphs, that the upper invariant
measure dominates a homogeneous product measure
with large density if the infection rate
$\lambda$ is sufficiently large.
As a consequence, this measure percolates if the corresponding
product measure percolates. We raise the question of whether domination
holds in the symmetric
case for all infinite graphs of bounded degree.
We study some asymmetric examples which we
feel shed some light on this question.
We next obtain necessary and sufficient conditions for
domination of a product measure for
``downward'' FKG measures. As a consequence of this
general result,
we show that the plus and minus states
for the Ising model on $Z^d$ dominate
the same set of product measures.
We show that this latter fact fails completely on the homogenous
3ary tree.
We also provide a different distinction between $Z^d$ and the homogenous 3ary tree
concerning stochastic domination and Ising models; while it is known
that the plus states for
different temperatures on $Z^d$ are never
stochastically ordered, on the homogenous 3ary tree,
almost the complete opposite is
the case. Next, we show that on $Z^d$, the set of product measures
which the plus state for the Ising model dominates is strictly increasing in the
temperature. Finally,
we obtain a necessary and sufficient condition for a finite
number of variables, which are both FKG
and exchangeable, to dominate a given product measure.

F. den Hollander, J. Steif, P. van der Wal
Bad Configurations for Random Walk
in Random Scenery and Related Subshifts,
Stochastic processes and their applications,
115, (2005), 12091232.
In this paper we consider an arbitrary irreducible random walk on $\Z^d$, $d \geq 1$,
with
i.i.d.\ increments, together with
an arbitrary i.i.d.\ random scenery. Walk and scenery are
assumed to be independent.
Random walk in random scenery (RWRS)
is the random process
where time is indexed by $\Z$, and at each unit of time both the step
taken by the walk and the
scenery value
at the site that is visited are registered.
Bad configurations for RWRS are the
discontinuity points of the conditional probability
distribution for the configuration at the
origin of time given the configuration at
all other times. We show that the set of bad
configurations is nonempty. We give a
complete description of this set and compute its
probability under the random scenery
measure. Depending on the type of random walk, this
probability may be zero or positive.
For simple symmetric random walk we get three
different types of behavior depending on whether
$d=1,2$, $d=3,4$ or $d \geq 5$. Our
classification is actually valid for a class of
subshifts having a certain determinative property,
of which RWRS is an example.
We also consider bad configurations w.r.t.\ a finite
time
interval (replacing the
origin) and obtain an almost complete generalization of our results.
Remarkably, this
extension turns out to be somewhat delicate.

E. Broman and J. Steif
Dynamical Stability of Percolation for Some Interacting Particle Systems
and $\epsilon$Stability,
Annals of Probability, 34, (2006), 539576.
In this paper we will investigate dynamic stability
of percolation for the stochastic
Ising model
and the contact process.
We also introduce the notion
of downwards and upwards $\epsilon$
stability which will be a key tool for our
analysis.

N. Gantert, M. Löwe and J. Steif
The voter model with antivoter bonds,
Annales Institut Henri Poincare, Probabilites et Statistiques,
41, (2005), 767780.
We study the voter model with both positive and negative bonds on
a general locally finite
connected infinite graph.
We obtain various results concerning ergodicity of the process.

F. den Hollander, M.S. Keane, J. Serafin and J. Steif
Weak Bernoullicity of Random Walk in Random Scenery,
Japanese Journal of Mathematics, 29, (2003), 389406.
In this paper, we consider an arbitrary irreducible random walk and
an arbitrary stationary
and ergodic random scenery on Z^d.
We find conditions on their distributions such that the
associated
random walk in random scenery is or is not weak Bernoulli. Our results
extend
an earlier classification for the special case in which the
individual scenery values are assumed
to be independent and identically
distributed random variables.

M. Keane and J. Steif
Finitary coding for the 1D T,T^{1}process with drift,
Annals of Probability, 31, (2003), 19791985.
We show that there is a finitary isomorphism from a finite
state i.i.d. process to the T,T^{1}
process associated
to 1d random walk with positive drift. This contrasts with
the situation for
simple symmetric random walk in any dimension,
where it cannot be a finitary factor of any
i.i.d. process
including in d > = 5 where it becomes weak Bernoulli.

R. Lyons and J. Steif
Stationary Determinantal Processes:
Phase Multiplicity, Bernoullicity, Entropy, and Domination,
Duke Mathematical Journal, 120, (2003), 515575.
We study a class of stationary processes indexed by Z^d that
are defined via minors of
ddimensional Toeplitz matrices.
We obtain necessary and sufficient conditions for
the
existence of a phase transition (phase multiplicity)
analogous to that which occurs in
statistical mechanics.
The absence of a phase transition is equivalent to the presence of a
strong
K property, a particular strengthening of the usual K
(Kolmogorov) property.
We
show that all of these processes are
Bernoulli shifts (isomorphic to i.i.d. processes in the
sense of ergodic
theory).
We obtain estimates of their entropies and
we relate these processes
via stochastic domination to product measures.

I. Benjamini, O. Häggström, Y. Peres and
J. Steif
Which properties of a random sequence are dynamically sensitive?,
Annals of Probability, 31, (2003), 134.
Consider a sequence of i.i.d. random variables, where each variable is
refreshed (i.e., replaced
by an independent variable
with the same law) independently, according to a Poisson clock.
At any fixed time t, the resulting sequence has the same
law as at time 0, but there can be
exceptional random times at
which certain almost sure properties of the time 0 sequence are
violated.
We prove that there are no such exceptional times for the law of large
numbers and
the law of the iterated logarithm,
so these laws are dynamically stable. However, there are
times
at which run lengths are exceptionally long, i.e., run tests are
dynamically sensitive. We
obtain a
multifractal analysis of exceptional times for run lengths
and for prediction. In particular,
starting from an i.i.d.
sequence of unbiased random bits, the random set of times t
where
alpha log_2(n) bits among the first n bits can be predicted
from their predecessors, has Hausdorff
dimension 1alpha a.s.
Finally, we consider simple random walk in the lattice Z^d, and prove that
transience is dynamically stable for d >= 5,
and dynamically sensitive for d=3,4.
Moreover, for
d=3,4, the nonempty
random set of exceptional times t where the walk
is recurrent, has
Hausdorff dimension (4d)/2 a.s.

R. Meester and J. Steif
Higherdimensional subshifts of finite type, factor maps
and measures of maximal entropy,
Pacific Mathematical Journal,
200, (2001), 497510.
We investigate factor maps of higherdimensional subshifts of finite
type. First we give higher
dimensional versions of well known onedimensional
results, including a characterization of
entropypreserving factor maps.
Next, we investigate what happens to the number of measures
of maximal
entropy under factor maps. We show that this number is preserved under
almost
invertible maps, but not in general under finite to one factor maps.
 J. Steif
The T,T^{1}process, finitary codings and weak Bernoulli ,
Israel Journal of Mathematics, 125, (2001), 2943.
We give an
elementary proof that the second coordinate (the scenery process)
of the T,T^{1}process
associated to any mean zero
i.i.d.
random walk on Z^d is not a finitary factor of an i.i.d. process.
In
particular, this yields an elementary proof that the basic
T,T^{1}process is not finitarily isomorphic to
a Bernoulli shift
(the stronger fact that it is not Bernoulli was proved by Kalikow).
This also provides
(using past work of den Hollander and the author) an
elementary example, namely the T,T^{1}process
in 5 dimensions,
of a process which is weak Bernoulli by not a finitary factor of
an i.i.d. process. An example
of such a process was given earlier
by del Junco and Rahe. The above
holds true for arbitrary stationary
recurrent random walks as well.
On the other hand, if the random walk is Bernoulli and transient, the
T,T^{1}process associated to it is also Bernoulli.
Finally, we show that finitary factors of i.i.d. processes
with finite expected coding volume satisfy certain notions of weak
Bernoulli in higher dimensions which have
been previously introduced
and studied in the literature. In particular, this yields
(using past work of van
den Berg and the author) the fact that the
Ising model is weak Bernoulli throughout the subcritical regime.
 O. Häggström, R. Schonmann and J. Steif
The Ising Model on Diluted Graphs and Strong Amenability,
Annals of Probability, 28, (2000), 11111137.
Say that a graph has persistent transition if the Ising model on the graph
can exhibit a phase transition
(nonuniqueness of Gibbs measures) in
the presence of a nonzero external field. We show that for
nonamenable graphs, for Bernoulli percolation with p close to 1,
all the infinite clusters have persistent
transition. On the other hand,
we show that for transitive amenable graphs, the infinite
clusters for any
stationary percolation do not have persistent
transition.
This extends a result of Georgii for the cubic
lattice. A
geometric consequence of this latter fact is that the
infinite clusters are strongly amenable
(i.e., their anchored Cheeger
constant is 0).
Finally we show that the critical temperature for the Ising
model with
no external field on the infinite clusters of Bernoulli percolation
with parameter p, on an
arbitrary bounded degree
graph, is a continuous function of p.

O. Häggström and J. Steif
ProppWilson algorithms and finitary codings for high noise Markov
random fields,
Combinatorics, Probability & Computing,
9, (2000), 425439.
In this paper, we combine two previous works, the first being by
the first author and K. Nelander, and the
second by
J. van den Berg and the second author, to show (1) that one can carry out
a ProppWilson exact
simulation for all Markov random fields
on Z^d satisfying a certain high noise assumption,
and (2) that all
such random fields are a finitary image of
a finite state i.i.d. process. (2) is a strengthening of the previously
known fact that such random fields are Bernoulli shifts.

R. Pemantle and J. Steif
Robust Phase Transitions for Heisenberg and Other Models
on General Trees,
Annals of Probability, 27, (1999), 876912.
We study several statistical mechanical models on a general tree.
Particular attention is devoted to the
spherical models, where the state
space is the ddimensional unit sphere
and the interactions are
proportional to
the cosines of the angles between neighboring spins. The phenomenon of
interest here
is the classification of phase transition
(nonuniqueness of the Gibbs state) according to whether it is
robust.
In many cases, including all of the spherical and Potts models,
occurrence of robust phase
transition is determined by the geometry
(branching number) of
the tree in a way that parallels the
situation with independent
percolation and usual phase transition for the Ising model. The critical
values
for robust phase transition for the spherical and Potts
models are also calculated exactly. In some cases,
such as the q >= 3 Potts
model, robust phase transition and usual phase transition do not coincide,
while
in other cases, such as the spherical models, we conjecture that robust
phase transition and usual phase
transition are equivalent. In addition,
we show that symmetry breaking is equivalent to the existence of a
phase
transition, a fact believed but not known for the rotor model on Z^2.

J. van den Berg and J. Steif
On the existence and nonexistence of finitary codings for a class
of random fields,
Annals of Probability, 27, (1999), 15011522.
We study the existence of finitary codings (also called finitary
homomorphisms or finitary factor maps) from a
finitevalued i.i.d. process
to certain random fields. For Markov random fields we show,
using ideas of Marton
and Shields,
that the presence of a phase transition is
an obstruction for the existence of the above coding:
this yields
a large class of Bernoulli shifts for which no such coding exists.
Conversely, we show that for the
stationary distribution of a monotone
exponentially ergodic probabilistic cellular automaton such a coding does
exist.
The construction of the coding is partially
inspired by the ProppWilson algorithm for exact simulation.
In particular, combining our results with
a theorem of Martinelli and Olivieri, we obtain the fact
that for the plus
state for the ferromagnetic Ising model on Z^d,
d >= 2, there is such a coding when the interaction
parameter
is below its critical value and
there is no such coding when the interaction
parameter is above its critical value.

J. Jonasson and J. Steif
Amenability and phase transition in the Ising model,
Journal of Theoretical Probability, 12, 1999, 549559.
We consider the Ising model with external field h and coupling constant
J on an infinite connected graph G with
uniformly bounded degree.
We prove that if G is nonamenable, then the Ising model exhibits phase
transition
for some h \neq 0 and some J<\infty.
On the other hand, if G is amenable and quasitransitive, then phase
transition
cannot occur for h \neq 0.
In particular, a group is nonamenable if and only if the Ising model on
one (all) of its
Cayley graphs exhibits a phase transition for some h \neq 0
and some J<\infty.
Last modified: Thu Aug 22 11:14:12 MET DST 2013