Johan Jonasson

Uniqueness of uniform random colorings of regular trees

A q-coloring of an infinite graph G is a homomorphism from G to the complete graph Kq on q vertices. A probability measure on the set of q-colorings of G is said to be a Gibbs measure for q-colorings of G for uniform activities if for every finite portion U of G and almost every q-coloring of G[-45 degree rule]U, the conditional distribution on the coloring of U given the coloring of G[-45 degree rule]U is uniform (on the set of colorings that are admissable when the coloring of the boundary of U is taken into account). In Brightwell and Winkler (2000), one studies q-colorings of the r+1-regular tree and among other things it is shown that if q[less-than-or-equals, slant]r+1 there are multiple such Gibbs measures, whereas when r is large enough and q[greater-or-equal, slanted]1.6296r there is a unique Gibbs measure. In this paper the gap is filled in: we show that for r[greater-or-equal, slanted]1000 one has uniqueness as soon as q[greater-or-equal, slanted]r+2. Computer calculations verify that the result is also true for 3[less-than-or-equals, slant]r

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

Amenability and Phase Transition in the Ising Model

{\mathbb{Z}}^d

On the Cover Time for Random Walks on Random Graphs

and, more generally, on transitive graphs. For iid percolation on

Dynamical models for circle covering: Brownian motion and Poisson updating

{\mathbb{Z}}^d

On Positive Random Objects

, 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.

Positive Influence and Negative Dependence

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≠0 and some J<∞. On the other hand, if G is amenable and quasi-transitive, then phase transition cannot occur for h≠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≠0 and some J<∞.

Rates of convergence for lamplighter processes

The cover time, C, for a simple random walk on a realization, GN, of G(N, p), the random graph on N vertices where each two vertices have an edge between them with probability p independently, is studied. The parameter p is allowed to decrease with N and p is written on the form f(N)/N, where it is assumed that f(N)≥c log N for some c>1 to asymptotically ensure connectedness of the graph. It is shown that if f(N) is of higher order than log N, then, with probability 1−o(1), (1−e)N log N≤E[C∣GN] ≤(1+e)N log N for any fixed e>0, whereas if f(N)=O(log N), there exists a constant a>0 such that, with probability 1−o(1), E[C∣GN] ≥(1+a)N log N. It is furthermore shown that if f(N) is of higher order than (log N)3 then Var(C∣GN)= o((N log N)2) with probability 1−o(1), so that with probability 1−o(1), the stronger statement that (1−e)N log N≤C≤(1+e)N log N holds.

Biased random-to-top shuffling

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 l is updated at rate l−α where α≥0 is a parameter. For the model with Brownian motions, a special case of our results is that if the length of the nth 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<α and c<α+1 instead. We also compute the Hausdorff dimension of the set of exceptional times for some of these questions.

Order Sampling Design with Prescribed Inclusion Probabilities

The idea of defining the expectation of a random variable as its integral with respect to a probability measure is extended to certain lattice-valued random objects and basic results of integration theory are generalized. Conditional expectation is defined and its properties are developed. Lattice valued martingales are also studied and convergence of sub- and supermartingales and the Optional Sampling Theorem are proved. A martingale proof of the Strong Law of Large Numbers is given. An extension of the lattice is also studied. Studies of some applications, such as on random compact convex sets in Rn and on random positive upper semicontinuous functions, are carried out, where the generalized integral is compared with the classical definition. The results are also extended to the case where the probability measure is replaced by a σ-finite measure.