Uniqueness and non-uniqueness in percolation theory

This paper is an up-to-date introduction to the problem of uniqueness versus non-uniqueness of infinite clusters for percolation on ${\mathbb{Z}}^d$ and, more generally, on transitive graphs. For iid percolation on ${\mathbb{Z}}^d$, uniqueness of the infinite cluster is a classical result, while on certain other transitive graphs uniqueness may fail. Key properties of the graphs in this context turn out to be amenability and nonamenability. The same problem is considered for certain dependent percolation models -- most prominently the Fortuin--Kasteleyn random-cluster model -- and in situations where the standard connectivity notion is replaced by entanglement or rigidity. So-called simultaneous uniqueness in couplings of percolation processes is also considered. Some of the main results are proved in detail, while for others the proofs are merely sketched, and for yet others they are omitted. Several open problems are discussed.Comment: Published at http://dx.doi.org/10.1214/154957806000000096 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Biased random-to-top shuffling

Recently Wilson [Ann. Appl. Probab. 14 (2004) 274--325] introduced an important new technique for lower bounding the mixing time of a Markov chain. In this paper we extend Wilson's technique to find lower bounds of the correct order for card shuffling Markov chains where at each time step a random card is picked and put at the top of the deck. Two classes of such shuffles are addressed, one where the probability that a given card is picked at a given time step depends on its identity, the so-called move-to-front scheme, and one where it depends on its position. For the move-to-front scheme, a test function that is a combination of several different eigenvectors of the transition matrix is used. A general method for finding and using such a test function, under a natural negative dependence condition, is introduced. It is shown that the correct order of the mixing time is given by the biased coupon collector's problem corresponding to the move-to-front scheme at hand. For the second class, a version of Wilson's technique for complex-valued eigenvalues/eigenvectors is used. Such variants were presented in [Random Walks and Geometry (2004) 515--532] and [Electron. Comm. Probab. 8 (2003) 77--85]. Here we present another such variant which seems to be the most natural one for this particular class of problems. To find the eigenvalues for the general case of the second class of problems is difficult, so we restrict attention to two special cases. In the first case the card that is moved to the top is picked uniformly at random from the bottom $k=k(n)=o(n)$ cards, and we find the lower bound $(n^3/(4\pi^2k(k-1)))\log n$. Via a coupling, an upper bound exceeding this by only a factor 4 is found. This generalizes Wilson's [Electron. Comm. Probab. 8 (2003) 77--85] result on the Rudvalis shuffle and Goel's [Ann. Appl. Probab. 16 (2006) 30--55] result on top-to-bottom shuffles. In the second case the card moved to the top is, with probability 1/2, the bottom card and with probability 1/2, the card at position $n-k$. Here the lower bound is again of order $(n^3/k^2)\log n$, but in this case this does not seem to be tight unless $k=O(1)$. What the correct order of mixing is in this case is an open question. We show that when $k=n/2$, it is at least $\Theta(n^2)$.Comment: Published at http://dx.doi.org/10.1214/10505160600000097 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Dynamical models for circle covering: Brownian motion and Poisson updating

We consider two dynamical variants of Dvoretzky's 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 $\alpha \ge0$ 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.Comment: Published in at http://dx.doi.org/10.1214/07-AOP340 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Multiplication of solutions for linear overdetermined systems of partial differential equations

A large family of linear, usually overdetermined, systems of partial differential equations that admit a multiplication of solutions, i.e, a bi-linear and commutative mapping on the solution space, is studied. This family of PDE's contains the Cauchy-Riemann equations and the cofactor pair systems, included as special cases. The multiplication provides a method for generating, in a pure algebraic way, large classes of non-trivial solutions that can be constructed by forming convergent power series of trivial solutions.Comment: 27 page

Explicit isoperimetric constants and phase transitions in the random-cluster model

The random-cluster model is a dependent percolation model that has applications in the study of Ising and Potts models. In this paper, several new results are obtained for the random-cluster model on nonamenable graphs with cluster parameter $q\geq 1$. Among these, the main ones are the absence of percolation for the free random-cluster measure at the critical value, and examples of planar regular graphs with regular dual where \pc^\f (q) > \pu^\w (q) for $q$ large enough. The latter follows from considerations of isoperimetric constants, and we give the first nontrivial explicit calculations of such constants. Such considerations are also used to prove non-robust phase transition for the Potts model on nonamenable regular graphs

Coupling and Bernoullicity in random-cluster and Potts models

An explicit coupling construction of random-cluster measures is presented. As one of the applications of the construction, the Potts model on amenable Cayley graphs is shown to exhibit at every temperature the mixing property known as Bernoullicity

Locational Determinants of Rural Non-agricultural Employment: Evidence From Brazil

By paying particular attention to the local economic context, this paper analyzes the determinants of non-agricultural employment and earnings in non-agricultural jobs. The empirical analysis is based on the Brazilian Demographic Census, allowing for disaggregated controls for the local economy. Education stands out as one of the key determinants of employment outcome and earnings potential. Failure to control for locational effects, however, can lead to biased estimation of the importance of individual and household-specific characteristics. The empirical results show that local market size and distance to population centers have a significant impact on non-agricultural employment prospects and earnings.Rural non-agricultural employment, economic geography, Latin America, Brazil

Visibility to infinity in the hyperbolic plane, despite obstacles

Suppose that $Z$ is a random closed subset of the hyperbolic plane \H^2, whose law is invariant under isometries of \H^2. We prove that if the probability that $Z$ contains a fixed ball of radius 1 is larger than some universal constant $p<1$, then there is positive probability that $Z$ contains (bi-infinite) lines. We then consider a family of random sets in \H^2 that satisfy some additional natural assumptions. An example of such a set is the covered region in the Poisson Boolean model. Let $f(r)$ be the probability that a line segment of length $r$ is contained in such a set $Z$. We show that if $f(r)$ decays fast enough, then there are almost surely no lines in $Z$. We also show that if the decay of $f(r)$ is not too fast, then there are almost surely lines in $Z$. In the case of the Poisson Boolean model with balls of fixed radius $R$ we characterize the critical intensity for the almost sure existence of lines in the covered region by an integral equation. We also determine when there are lines in the complement of a Poisson process on the Grassmannian of lines in \H^2