A stochastic lipid structured model for macrophage dynamics in atherosclerotic plaques.

We propose to model certain aspects of the dynamics of a macrophage that moves randomly in a one dimensional space in arterial wall tissue and grows by accumulating localized lipid particles, thus reducing its motility. This phenomenon has been observed in the context of atherosclerotic plaque formation. For this purpose, we use a system of stochastic differential equations satisfied by the position and diffusion coefficient of a Brownian particle whose diffusion coefficient is modified at each visit to the origin and with a dumping coefficient. The novelty of the model, with respect to Bénichou et al. (Phys Rev E 85(2):021137, 2012), Meunier et al. (Acta Appl Math 161:107-126, 2019), is to include offloading of lipids through the dumping term. We find explicit necessary and sufficient conditions for macrophage trapping in the locally enriched region.

On the tail of the branching random walk local time.

Consider a critical branching random walk on Z d , d ≥ 1 , started with a single particle at the origin, and let L(x) be the total number of particles that ever visit a vertex x. We study the tail of L(x) under suitable conditions on the offspring distribution. In particular, our results hold if the offspring distribution has an exponential moment.

The bead process for beta ensembles.

The bead process introduced by Boutillier is a countable interlacing of the Sine 2 point processes. We construct the bead process for general Sine ß processes as an infinite dimensional Markov chain whose transition mechanism is explicitly described. We show that this process is the microscopic scaling limit in the bulk of the Hermite ß corner process introduced by Gorin and Shkolnikov, generalizing the process of the minors of the Gaussian Unitary and Orthogonal Ensembles. In order to prove our results, we use bounds on the variance of the point counting of the circular and the Gaussian beta ensembles, proven in a companion paper (Najnudel and Virág in Some estimates on the point counting of the Circular and the Gaussian Beta Ensemble, 2019).

Favorite Sites of a Persistent Random Walk.

We consider favorite (i.e., most visited) sites of a symmetric persistent random walk on ℤ , a discrete-time process typified by the correlation of its directional history. We show that the cardinality of the set of favorite sites is eventually at most three. This is a generalization of a result by Tóth for a simple random walk, used to partially prove a longstanding conjecture by Erdos and Róvósz. The original conjecture asserting that for the simple random walk on integers the cardinality of the set of favorite sites is eventually at most two was recently disproved by Ding and Shen.