Transience and recurrence of random walks on percolation clusters in an ultrametric space

We study existence of percolation in the hierarchical group of order $N$, which is an ultrametric space, and transience and recurrence of random walks on the percolation clusters. The connection probability on the hierarchical group for two points separated by distance $k$ is of the form $c_k/N^{k(1+\delta)}, \delta>-1$, with $c_k=C_0+C_1\log k+C_2k^\alpha$, non-negative constants $C_0, C_1, C_2$, and $\alpha>0$. Percolation was proved in Dawson and Gorostiza (2013) for $\delta0$, with $\alpha>2$. In this paper we improve the result for the critical case by showing percolation for $\alpha>0$. We use a renormalization method of the type in the previous paper in a new way which is more intrinsic to the model. The proof involves ultrametric random graphs (described in the Introduction). The results for simple (nearest neighbour) random walks on the percolation clusters are: in the case $\delta<1$ the walk is transient, and in the critical case $\delta=1, C_2>0,\alpha>0$, there exists a critical $\alpha_c\in(0,\infty)$ such that the walk is recurrent for $\alpha<\alpha_c$ and transient for $\alpha>\alpha_c$. The proofs involve graph diameters, path lengths, and electric circuit theory. Some comparisons are made with behaviours of random walks on long-range percolation clusters in the one-dimensional Euclidean lattice.Comment: 27 page

Hierarchical equilibria of branching populations

The objective of this paper is the study of the equilibrium behavior of a population on the hierarchical group $\Omega_N$ consisting of families of individuals undergoing critical branching random walk and in addition these families also develop according to a critical branching process. Strong transience of the random walk guarantees existence of an equilibrium for this two-level branching system. In the limit $N\to\infty$ (called the hierarchical mean field limit), the equilibrium aggregated populations in a nested sequence of balls $B^{(N)}_\ell$ of hierarchical radius $\ell$ converge to a backward Markov chain on $\mathbb{R_+}$. This limiting Markov chain can be explicitly represented in terms of a cascade of subordinators which in turn makes possible a description of the genealogy of the population.Comment: 62 page

Higgs Boson Production with Bottom Quarks at Hadron Colliders

We present results for the production cross section of a Higgs Boson with a pair of bottom/anti-bottom quarks, including next-to-leading order (NLO) QCD corrections.Comment: 3 pages, 2 figures, uses ws-ijmpa.cls. Talk given by C.B. Jackson at the Meeting of the Division of Particles and Fields (DPF2004) in Riverside, CA, August 26-31, 200

Theoretical progress for the associated production of a Higgs boson with heavy quarks at hadron colliders

The production of a Higgs boson in association with a pair of top-antitop or bottom-antibottom quarks plays a very important role at both the Tevatron and the Large Hadron Collider. The theoretical prediction of the corresponding cross sections has been improved by including the complete next-to-leading order QCD corrections. After a brief introduction, we review the results obtained for both the Tevatron and the Large Hadron Collider.Comment: 3 pages, 6 figures, uses svjour.cls. Talk given by L. Reina at the HEP2003 Europhysics Conference in Aachen, Germany (EPS 2003), July 17-23, 200

A simple encoding of a quantum circuit amplitude as a matrix permanent

A simple construction is presented which allows computing the transition amplitude of a quantum circuit to be encoded as computing the permanent of a matrix which is of size proportional to the number of quantum gates in the circuit. This opens up some interesting classical monte-carlo algorithms for approximating quantum circuits.Comment: 6 figure

Peripheral visual response time to colored stimuli imaged on the horizontal meridian

Two male observers were administered a binocular visual response time task to small (45 min arc), flashed, photopic stimuli at four dominant wavelengths (632 nm red; 583 nm yellow; 526 nm green; 464 nm blue) imaged across the horizontal retinal meridian. The stimuli were imaged at 10 deg arc intervals from 80 deg left to 90 deg right of fixation. Testing followed either prior light adaptation or prior dark adaptation. Results indicated that mean response time (RT) varies with stimulus color. RT is faster to yellow than to blue and green and slowest to red. In general, mean RT was found to increase from fovea to periphery for all four colors, with the curve for red stimuli exhibiting the most rapid positive acceleration with increasing angular eccentricity from the fovea. The shape of the RT distribution across the retina was also found to depend upon the state of light or dark adaptation. The findings are related to previous RT research and are discussed in terms of optimizing the color and position of colored displays on instrument panels