Brownian Bridge and Self-Avoiding Random Walk

We establish the Brownian bridge asymptotics for a scaled self-avoiding walk conditioned on arriving to a far away point $n \vec{a}$ for $\vec{a}$ in $Z^d$, as $n$ increases to infinity

Brownian Bridge Asymptotics for the Subcritical Bernoulli Bond Percolation

For the d-dimensional model of a subcritical bond percolation (p<p_c) and a point \vec{a} in Z^d, we prove that a cluster conditioned on connecting points (0,...,0) and n\vec{a} if scaled by 1/(n|vec{a}|) along \vec{a} and by 1/sqrt{n} in the orthogonal direction converges asymptotically to Time x (d-1)-dimensional Brownian Bridge.Comment: 18 pages, no figures, LaTe

Exclusion Processes with Multiple Interactions

We introduce the mathematical theory of the particle systems that interact via permutations, where the transition rates are assigned not to the jumps from a site to a site, but to the permutations themselves. This permutation processes can be viewed as a generalization of the symmetric exclusion processes, where particles interact via transpositions. The duality and coupling techniques for the processes are described, the needed conditions for them to apply are established. The stationary distributions of the permutation processes are explored for translation invariant cases.Comment: 23 page

Orthogonality and probability: mixing times

We produce the first example of bounding total variation distance to stationarity and estimating mixing times via orthogonal polynomials diagonalization of discrete reversible Markov chains, the Karlin-McGregor approach

Multi-particle processes with reinforcements

We consider a multi-particle generalization of linear edge-reinforced random walk (ERRW). We observe that in absence of exchangeability, new techniques are needed in order to study the multi-particle model. We describe an unusual coupling construction associated with the two-point edge-reinforced process on Z and prove a form of recurrence: the two particles meet infinitely often a.s.Comment: 12 page

Linear speed large deviations for percolation clusters

Let C_n be the origin-containing cluster in subcritical percolation on the lattice (1/n) Z^d, viewed as a random variable in the space Omega of compact, connected, origin-containing subsets of R^d, endowed with the Hausdorff metric delta. When d >= 2, and Gamma is any open subset of Omega, we prove: lim_{n \to \infty}(1/n) \log P(C_n \in \Gamma) = -\inf_{S \in \Gamma} \lambda(S) where lambda(S) is the one-dimensional Hausdorff measure of S defined using the {\em correlation norm}: ||u|| := \lim_{n \to \infty} - \frac{1}{n} \log P (u_n \in C_n) where u_n is u rounded to the nearest element of (1/n)Z^d. Given points a^1, >..., a^k in R^d, there are finitely many correlation-norm Steiner trees spanning these points and the origin. We show that if the C_n are each conditioned to contain the points a^1_n,..., a^k_n, then the probability that C_n fails to approximate one of these trees decays exponentially in n.Comment: five page

Horton Law in Self-Similar Trees

Self-similarity of random trees is related to the operation of pruning. Pruning $R$ cuts the leaves and their parental edges and removes the resulting chains of degree-two nodes from a finite tree. A Horton-Strahler order of a vertex $v$ and its parental edge is defined as the minimal number of prunings necessary to eliminate the subtree rooted at $v$. A branch is a group of neighboring vertices and edges of the same order. The Horton numbers $N_k[K]$ and $N_{ij}[K]$ are defined as the expected number of branches of order $k$, and the expected number of order-$i$ branches that merged order-$j$ branches, $j>i$, respectively, in a finite tree of order $K$. The Tokunaga coefficients are defined as $T_{ij}[K]=N_{ij}[K]/N_j[K]$. The pruning decreases the orders of tree vertices by unity. A rooted full binary tree is said to be mean-self-similar if its Tokunaga coefficients are invariant with respect to pruning: $T_k:=T_{i,i+k}[K]$. We show that for self-similar trees, the condition $\limsup(T_k)^{1/k}<\infty$ is necessary and sufficient for the existence of the strong Horton law: $N_k[K]/N_1[K] \rightarrow R^{1-k}$, as $K \rightarrow \infty$ for some $R>0$ and every $k\geq 1$. This work is a step toward providing rigorous foundations for the Horton law that, being omnipresent in natural branching systems, has escaped so far a formal explanation

Horton self-similarity of Kingman's coalescent tree

The paper establishes a weak version of Horton self-similarity for a tree representation of Kingman's coalescent process. The proof is based on a Smoluchowski-type system of ordinary differential equations for the number of branches of a given Horton-Strahler order in a tree that represents Kingman's N-coalescent process with a constant kernel, in a hydrodynamic limit. We also demonstrate a close connection between the combinatorial Kingman's tree and the combinatorial level set tree of a white noise, which implies Horton self-similarity for the latter

Mixing times via super-fast coupling

We provide a coupling proof that the transposition shuffle on a deck of n cards is mixing of rate Cn(log{n}) with a moderate constant, C. This rate was determined by Diaconis and Shahshahani, but the question of a natural probabilistic coupling proof has been missing, and questions of its existence have been raised. The proof, and indeed any proof, requires that we enlarge the methodology of coupling to include intuitive but non-adapted coupling rules, because a typical Markovian coupling is incapable of resolving finer questions of rates

Random Self-Similar Trees: A mathematical theory of Horton laws

The Horton laws originated in hydrology with a 1945 paper by Robert E. Horton, and for a long time remained a purely empirical finding. Ubiquitous in hierarchical branching systems, the Horton laws have been rediscovered in many disciplines ranging from geomorphology to genetics to computer science. Attempts to build a mathematical foundation behind the Horton laws during the 1990s revealed their close connection to the operation of pruning -- erasing a tree from the leaves down to the root. This survey synthesizes recent results on invariances and self-similarities of tree measures under various forms of pruning. We argue that pruning is an indispensable instrument for describing branching structures and representing a variety of coalescent and annihilation dynamics. The Horton laws appear as a characteristic imprint of self-similarity, which settles some questions prompted by geophysical data.Comment: 208 pages, 50 figure