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We consider a one-dimensional traffic model with a slow-to-start rule. The initial position of the cars in $mathbb R$ is a Poisson process of parameter $lambda$. Cars have speed 0 or 1 and travel in the same direction. At time zero the speed of all cars is 0; each car waits an exponential time to switch speed from $0$ to $1$ and stops when it collides with a stopped car. When the car is no longer blocked, it waits a new exponential time to assume speed one, and so on. We study the emergence of condensation for the saturated regime $lambda>1$ and the critical regime $lambda=1$, showing that in both regimes all cars collide infinitely often and each car has asymptotic mean velocity $1/lambda$. In the saturated regime the moving cars form a point process whose intensity tends to 1. The remaining cars condensate in a set of points whose intensity tends to zero as $1/sqrt t$. We study the scaling limit of the traffic jam evolution in terms of a collection of coalescing Brownian motions.
The dynamics of a population undergoing selection is a central topic in evolutionary biology. This question is particularly intriguing in the case where selective forces act in opposing directions at two population scales. For example, a fast-replica
We establish the existence of free energy limits for several combinatorial models on Erd{o}s-R{e}nyi graph $mathbb {G}(N,lfloor cNrfloor)$ and random $r$-regular graph $mathbb {G}(N,r)$. For a variety of models, including independent sets, MAX-CUT, c
Let $K_n$ be the convex hull of i.i.d. random variables distributed according to the standard normal distribution on $R^d$. We establish variance asymptotics as $n to infty$ for the re-scaled intrinsic volumes and $k$-face functionals of $K_n$, $k in
Let K be a convex set in R d and let K $lambda$ be the convex hull of a homogeneous Poisson point process P $lambda$ of intensity $lambda$ on K. When K is a simple polytope, we establish scaling limits as $lambda$ $rightarrow$ $infty$ for the boundar
We consider various asymptotic scaling limits $Ntoinfty$ for the $2N$ complex eigenvalues of non-Hermitian random matrices in the symmetry class of the symplectic Ginibre ensemble. These are known to be integrable, forming Pfaffian point processes, a