The style of mathematical models known to probabilists as Interacting Particle Systems and exemplified by the Voter, Exclusion and Contact processes have found use in many academic disciplines. In many such disciplines the underlying conceptual picture is of a social network, where individuals meet pairwise and update their "state" (opinion, activity etc) in a way depending on the two previous states. This picture motivates a precise general setup we call Finite Markov Information Exchange (FMIE) processes. We briefly describe a few less familiar models (Averaging, Compulsive Gambler, Deference, Fashionista) suggested by the social network picture, as well as a few familiar ones. © 2013 ISI/BS.

On a large finite connected graph let edges e become “open” at independent random Exponential times of arbitrary rates we. Under minimal assumptions, the time at which a giant component starts to emerge is weakly concentrated around its mean.

For random walks on finite graphs, we record some equalities, inequalities and limit theorems (as the size of graph tends to infinity) which hold for vertex-transitive graphs but not for general regular graphs. The main result is a sharp condition for asymptotic exponentiality of the hitting time to a single vertex. Another result is a lower bound for the coefficient of variation of hitting times. Proofs exploit the complete monotonicity properties of the associated continuous-time walk. © 1989, Cambridge Philosophical Society. All rights reserved.

We show that, in the usual probabilistic model for the random assignment problem, the optimal cost tends to a limit constant in probability and in expectation. The method involves construction of an infinite limit structure, in terms of which the limit constant is defined. But we cannot improve on the known numerical bounds for the limit. © 1992 Springer-Verlag.

Component sizes in the usual random graph process are a special case of the Marcus-Lushnikov process discussed in the scientific literature, so it is natural to ask how theory surrounding emergence of the giant component generalizes to the Marcus-Lushnikov process. Essentially no rigorous results are known; we make a start by proving a weak result, but our main purpose is to draw this topic to the attention of random graph theorists. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12, 179-196, 1998.

Jim Pitman was born in June 1949, received a Ph.D. in 1974 from the University of Sheffield with advisor Terry Speed, and since 1979 has been in the U.C. Berkeley Statistics department. He is known for research on many topics within probability, in particular for a long-running collaboration with Marc Yor on distributional properties of Brownian motion, and for his influential lecture notes Combinatorial Stochastic Processes. The following conversation took place at his home in December 2017 and February 2018.

Consider routing traffic on the N × N torus, simultaneously between all source-destination pairs, to minimize the cost ∑ec(e)f 2(e). where f(e) is the volume of flow across edge e and the c(e) form an i.i.d. random environment. We prove existence of a rescaled N → ∞ limit constant for minimum cost, by comparison with an appropriate analogous problem about minimum-cost flows across a M × M subsquare of the lattice. © Institute of Mathematical Statistics, 2007.

In a simple model for sports, the probability A beats B is a specified function of their difference in strength. One might think this would be a staple topic in Applied Probability textbooks (like the Galton-Watson branching process model, for instance) but it is curiously absent. Our first purpose is to point out that the model suggests a wide range of questions, suitable for "undergraduate research" via simulation but also challenging as professional research. Our second, more specific, purpose concerns Elo-type rating algorithms for tracking changing strengths. There has been little foundational research on their accuracy, despite a much-copied "30 matches suffice" claim, which our simulation study casts doubt upon.