Point-to-line last passage percolation and the invariant measure of a system of reflecting Brownian motions

This paper proves an equality in law between the invariant measure of a reflected system of Brownian motions and a vector of point-to-line last passage percolation times in a discrete random environment. A consequence describes the distribution of the all-time supremum of Dyson Brownian motion with drift. A finite temperature version relates the point-to-line partition functions of two directed polymers, with an inverse-gamma and a Brownian environment, and generalises Dufresne’s identity. Our proof introduces an interacting system of Brownian motions with an invariant measure given by a field of point-to-line log partition functions for the log-gamma polymer

Petrography, The Tar Sands Paradise, and the Medium of Modernity

This article engages with the artistic practice of petrography, the art of creating photographic images through the action of sunlight upon bitumen, the heavy-oil material that is the source of the petroleum in the Athabasca tar sands. It presents several examples of petrographs that document the process of industrial bitumen mining itself. Further, it theorizes the ways in which both the process of producing petrographs and the act of engaging with them as a viewer require a degree of collaboration normally absent from our consumption of petroleum as the medium of modernity. A key argument of the paper is the reconfiguration of bitumen as a medicine in Cree/Métis contexts, which leads to an alternative Indigenous idea of the petro-medium as an active, relational substance with its own potential agency

Probabilistic study of a dynamical system

This paper investigates the relation between a branching process and a non-linear dynamical system in C2. This idea has previously been fruitful in many investigations, including that of the FKPP equation by McKean, Neveu, Bramson, and others. Our concerns here are somewhat different from those in other work: we wish to elucidate those features of the dynamical system which correspond to the long-term behaviour of the random process. In particular, we are interested in how the dimension of the global attractor corresponds to that of the tail {sigma}-algebra of the process. The Poincaré–Dulac operator which (locally) intertwines the non-linear system with its linearization may sometimes be exhibited as a Fourier–Laplace transform of tail-measurable random variables; but things change markedly when parameters cross values giving the ‘primary resonance’ in the Poincaré–Dulac sense. Probability proves effective in establishing global properties amongst which is a clear description of the global convergence to the attractor. Several of our probabilistic results are analogues of ones obtained by Kesten and Stigum, and by Athreya and Ney, for discrete branching processes. Our simpler context allows the use of Itô calculus. Because the paper bridges two subjects, dynamical-system theory and probability theory, we take considerable care with the exposition of both aspects. For probabilist readers, we provide a brief guide to Poincaré–Dulac theory; and we take the view that in a paper which we hope will be read by analysts, it would be wrong to fudge any details of rigour in our probabilistic arguments