arXiv subjects

Johan Jonasson

Publications and source records attributed to Johan Jonasson.

25 records · Page 2Linked to original sources

arXiv2008

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 $\ell^{-α}$ where $α\ge0$ 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 $c<α$ and $c<α+1$ instead. We also compute the Hausdorff dimension of the set of exceptional times for some of these questions.

math.PR↗

arXiv2006

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.

math.PR↗

arXiv2006

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 $(n^3/(4π^2k(k-1)))\log n$. 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 $Θ(n^2)$.

math.PR↗

arXiv2006

The overhand shuffle mixes in $Θ(n^2\log n)$ steps

The overhand shuffle is one of the ``real'' card shuffling methods in the sense that some people actually use it to mix a deck of cards. A mathematical model was constructed and analyzed by Pemantle [J. Theoret. Probab. 2 (1989) 37--49] who showed that the mixing time with respect to variation distance is at least of order $n^2$ and at most of order $n^2\log n$. In this paper we use an extension of a lemma of Wilson [Ann. Appl. Probab. 14 (2004) 274--325] to establish a lower bound of order $n^2\log n$, thereby showing that $n^2\log n$ is indeed the correct order of the mixing time. It is our hope that the extension of Wilson's lemma will prove useful also in other situations; it is demonstrated how it may be used to give a simplified proof of the $Θ(n^3\log n)$ lower bound of Wilson [Electron. Comm. Probab. 8 (2003) 77--85] for the Rudvalis shuffle.

math.PR↗

arXiv2001

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.

math.PR↗

arXiv2001

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.

math.PR↗

arXiv2000

On the cover time of planar graphs

The cover time of a finite connected graph is the expected number of steps needed for a simple random walk on the graph to visit all the vertices. It is known that the cover time on any n-vertex, connected graph is at least (1+o(1)) n log(n) and at most (1+o(1))(4/27)n^3. This paper proves that for bounded-degree planar graphs the cover time is at least c n(log n)^2, and at most 6n^2, where c is a positive constant depending only on the maximal degree of the graph. The lower bound is established via use of circle packings.

math.PR↗