Power law Polya's urn and fractional Brownian motion (2009)
Hammond, Alan, Sheffield, Scott
We introduce a natural family of random walks on the set of integers that scale to fractional Brownian motion. The increments X_n have the property that given {X_k: k < n}, the conditional law of X_n...
Coagulation, diffusion and the continuous Smoluchowski equation (2008)
Yaghouti, Mohammad Reza, Rezakhanlou, Fraydoun, Hammond, Alan
The Smoluchowski equation is a system of partial differential equations modelling the diffusion and binary coagulation of a large collection of tiny particles. The mass parameter may be indexed...
Monotone loop models and rational resonance (2008)
Hammond, Alan, Kenyon, Richard
Let $T_{n,m}=\mathbb Z_n\times\mathbb Z_m$, and define a random mapping $\phi\colon T_{n,m}\to T_{n,m}$ by $\phi(x,y)=(x+1,y)$ or $(x,y+1)$ independently over $x$ and $y$ and with equal probability....
Biased random walks on a Galton-Watson tree with leaves (2007)
Arous, Gérard Ben, Fribergh, Alexander, Gantert, Nina, Hammond, Alan
We consider a biased random walk $X_n$ on a Galton-Watson tree with leaves in the sub-ballistic regime. We prove that there exists an explicit constant $\gamma= \gamma(\beta) \in (0,1)$, depending on...
Biased random walks on a Galton-Watson tree with leaves (2007)
Ben Arous, Gérard, Fribergh, Alexander, Gantert, Nina, Hammond, Alan
We consider a biased random walk $X_n$ on a Galton-Watson tree with leaves in the sub-ballistic regime. We prove that there exists an explicit constant $\gamma= \gamma(\beta) \in (0,1)$, depending on...
Biased random walks on a Galton-Watson tree with leaves (2007)
Ben Arous, Gérard, Fribergh, Alexander, Gantert, Nina, Hammond, Alan
We consider a biased random walk $X_n$ on a Galton-Watson tree with leaves in the sub-ballistic regime. We prove that there exists an explicit constant $\gamma= \gamma(\beta) \in (0,1)$, depending on...
Moment bounds for the Smoluchowski equation and their consequences (2006)
Hammond, Alan, Rezakhanlou, Fraydoun
We prove uniform bounds on moments X_a = \sum_{m}{m^a f_m(x,t)} of the Smoluchowski coagulation equations with diffusion, valid in any dimension. If the collision propensities \alpha(n,m) of mass n...
Greedy lattice animals: Geometry and criticality (2006)
Assign to each site of the integer lattice ℤd a real score, sampled according to the same distribution F, independently of the choices made at all other sites. A lattice animal is a finite...
The kinetic limit of a system of coagulating planar Brownian particles (2005)
Hammond, Alan, Rezakhanlou, Fraydoun
We study a model of mass-bearing coagulating planar Brownian particles. Coagulation is prone to occur when two particles become within a distance of order $\epsilon$. We assume that the initial...
Critical Exponents in Percolation via Lattice Animals (2005)
Hammond, Alan; U.C. Berkeley, USA; Alanmh@stat.berkeley.edu
We examine the percolation model on $mathbb{Z}^d$ by an approach involving lattice animals and their surface-area-to-volume ratio. For $beta in [0,2(d-1))$, let $f(beta)$ be the asymptotic...
Critical Exponents in Percolation via Lattice Animals (2005)
Hammond, Alan; U.C. Berkeley, USA; Alanmh@stat.berkeley.edu
We examine the percolation model on $mathbb{Z}^d$ by an approach involving lattice animals and their surface-area-to-volume ratio. For $beta in [0,2(d-1))$, let $f(beta)$ be the asymptotic...
Fluctuation of planar Brownian loop capturing large area (2004)
We consider a planar Brownian loop $B$ that is run for a time $T$ and conditioned on the event that its range encloses the unusually high area of $\pi T^2$, with $T$ being large. We study the...
Greedy lattice animals: geometry and criticality (with an Appendix) (2004)
Assign to each site of the integer lattice $\Zd$ a real score, sampled according to the same distribution $F$, independently of the choices made at all other sites. A lattice animal is a finite...
The kinetic limit of a system of coagulating Brownian particles (2004)
Hammond, Alan, Rezakhanlou, Fraydoun
We consider a random model of diffusion and coagulation. A large number of small particles are randomly scattered at an initial time. Each particle has some integer mass and moves in a Brownian...
Percolation and lattice animals: exponent relations, and conditions for $\theta(p_c)=0$ (2004)
We examine the percolation model in $\mathbb{Z}^d$ by an approach involving lattice animals, in which their relevant characteristic is surface-area-to-volume ratio. Two critical exponents are...
A lattice animal approach to percolation (2004)
We examine the percolation model on $\mathbb{Z}^d$ by an approach involving lattice animals and their surface-area-to-volume ratio. For $\beta \in [0,2(d-1))$, let $f(\beta)$ be the asymptotic...