J. Ben Hough, Graduate Student

• Resume available upon request.
• Papers on the Mathematics ArXiv.
• Research supported in part by NSF grants: #0104073 and #0244479.

• Probability theory.
• Discrete and continuous time stochastic processes.
• Applications to physics, computer science and finance.
• I am a PhD student of Yuval Peres.

• Large deviations for the zero set of an analytic function with diffusing coefficients.  (J. Ben Hough).  Preprint.

The "hole probability" that the zero set of the time dependent planar Gaussian analytic function

where a_n(t) are i.i.d. complex valued Ornstein-Uhlenbeck processes, does not intersect a disk of radius R for all 0≤t≤T decays like exp(-T exp(cR²)). This result sharply differentiates the zero set of f from a number of canonical evolving planar point processes. For example, the hole probability of the perturbed lattice model {π^1/2 (m,n) + c x_m,n: m,n integers} where x_m,n are i.i.d. Ornstein-Uhlenbeck processes decays like exp(-cTR⁴). This stark contrast is also present in the "overcrowding probability" that a disk of radius R contains at least N zeros for all 0≤t≤T.

Figure 1:  The zero set of f( ,t) (left) and the perturbed lattice model, conditioned to have a hole of radius 5.

• Determinantal Processes and Independence.  (J. Ben Hough, Manjunath Krishnapur, Yuval Peres and Bálint Virág).  Probability Surveys, 3, (2006), 206--229.

We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees).  They have the striking property that the number of points in a region D is a sum of independent Bernoulli random variables, with parameters which are eigenvalues of the relevant operator on L²(D).  Moreover, any determinantal process can be represented as a mixture of determinantal projection processes.  We give a simple explanation for these known facts, and establish analogous representations for permanental processes, with geometric variables replacing the Bernoulli variables.  These representations lead to simple proofs of existence criteria and central limit theorems, and unify known results on the distribution of absolute values in certain processes with radially symmetric distributions.

Figure 2:  Samples of translation invariant point processes in the plane: Poisson (left), determinantal (center) and permanental.
Determinantal processes exhibit repulsion, while permanental processes exhibit clumping. 

• An LIL for cover times of disks by planar random walk and Wiener sausage. (J. Ben Hough and Yuval Peres).  Trans. Amer. Math. Soc., to appear.

  Let R_n be the radius of the largest disk covered after n steps of a simple random walk. We prove that almost surely limsup_{n ® ¥}(log R_n)² /(log n log₃ n) = 1/4, where log₃ denotes 3 iterations of the log function. This is motivated by a question of Erdős and Taylor. We also obtain the analogous result for the Wiener sausage, refining a result of Meyre and Werner.

   

• Asymptotic Behavior of Random Heaps. (J. Ben Hough).  Random Structures and Algorithm, 27 (2005), no 4, 476--489.

  A random heap consists of a collection of particles with "+" and "-" charges distributed over the lattice Z_m × Z⁺ , where Z_m denotes the integers modulo m. It evolves in discrete time increments by randomly selecting a new particle with "+" or "-" charge equally likely, and then dropping it uniformly over the m columns of the lattice. When a particle is dropped in a column, it falls as far as possible subject to the condition that it cannot fall past a piece lying in it's own or an adjacent column, i.e. the particles have "sticky corners". A typical random heap is shown below:

  

  Figure 2: A random heap.
  
  A piece in a random heap is called a roof element if there are no pieces lying above it in its own column or the two adjacent columns (see the shaded pieces above). The size of the roof is an important feature of the heap, because it controls the rate of growth. When the roof is large, there is a better chance that the next piece dropped will annihilate one of the roof elements and the heap will diminish in size. It is easy to prove that if one only drops "+" charges then the expected size of the roof will be exactly m/3 as time tends to infinity, so it has been conjectured that the same asymptotic behavior should also be true if one allows "-" charges to be dropped as well. However, the above paper proves that this is not the case. There exists e > 0 so that the long term expected size of the roof is less than (1-e)m/3 for any m > 3.

Talks

• Large deviations for analytic functions with diffusing coefficients, Talk given at Stanford probability seminar.
• Counting Points in Determinantal and Permanental Processes I, Talk given at U.C. Davis mathematical physics/probability seminar.

Other works

• The Spectral Gap, lecture given at U.C. Berkeley for a course by Jim Pitman.
• Hall's Matching Theorem, presentation given at U.C. Berkeley for a course by Yuval Peres.
• Polarization of High Energy Neutrons in Proton-Nucleus Scattering, senior thesis in partial fulfilment of physics B.S. at M.I.T. Results are presented from a polarization measurement conducted at Los Alamos National Laboratory.
• The Path Integral Approach to Quantum Mechanics, paper written at M.I.T. for a course by Krishna Rajagopal.
• Millikan's Oil Drop Experiment, paper written while at M.I.T., Navier-Stoke's equations are discussed.

Other links

• U.C. Berkeley Department of Mathematics.
• U.C. Berkeley Probability
• U.C. Berkeley probability seminar

• Math 16B, Summer 2003