TY - JOUR
AB - We consider the totally asymmetric simple exclusion process (TASEP) with non-random initial condition having density ρ on ℤ− and λ on ℤ+, and a second class particle initially at the origin. For ρ<λ, there is a shock and the second class particle moves with speed 1−λ−ρ. For large time t, we show that the position of the second class particle fluctuates on a t1/3 scale and determine its limiting law. We also obtain the limiting distribution of the number of steps made by the second class particle until time t.
AU - Ferrari, Patrick
AU - Ghosal, Promit
AU - Nejjar, Peter
ID - 72
IS - 3
JF - Annales de l'institut Henri Poincare (B) Probability and Statistics
SN - 02460203
TI - Limit law of a second class particle in TASEP with non-random initial condition
VL - 55
ER -
TY - CONF
AB - We study edge asymptotics of poissonized Plancherel-type measures on skew Young diagrams (integer partitions). These measures can be seen as generalizations of those studied by Baik--Deift--Johansson and Baik--Rains in resolving Ulam's problem on longest increasing subsequences of random permutations and the last passage percolation (corner growth) discrete versions thereof. Moreover they interpolate between said measures and the uniform measure on partitions. In the new KPZ-like 1/3 exponent edge scaling limit with logarithmic corrections, we find new probability distributions generalizing the classical Tracy--Widom GUE, GOE and GSE distributions from the theory of random matrices.
AU - Betea, Dan
AU - Bouttier, Jérémie
AU - Nejjar, Peter
AU - Vuletíc, Mirjana
ID - 8175
T2 - Proceedings on the 31st International Conference on Formal Power Series and Algebraic Combinatorics
TI - New edge asymptotics of skew Young diagrams via free boundaries
ER -
TY - JOUR
AB - We investigate the free boundary Schur process, a variant of the Schur process introduced by Okounkov and Reshetikhin, where we allow the first and the last partitions to be arbitrary (instead of empty in the original setting). The pfaffian Schur process, previously studied by several authors, is recovered when just one of the boundary partitions is left free. We compute the correlation functions of the process in all generality via the free fermion formalism, which we extend with the thorough treatment of “free boundary states.” For the case of one free boundary, our approach yields a new proof that the process is pfaffian. For the case of two free boundaries, we find that the process is not pfaffian, but a closely related process is. We also study three different applications of the Schur process with one free boundary: fluctuations of symmetrized last passage percolation models, limit shapes and processes for symmetric plane partitions and for plane overpartitions.
AU - Betea, Dan
AU - Bouttier, Jeremie
AU - Nejjar, Peter
AU - Vuletic, Mirjana
ID - 556
IS - 12
JF - Annales Henri Poincare
SN - 14240637
TI - The free boundary Schur process and applications I
VL - 19
ER -
TY - JOUR
AB - We consider the totally asymmetric simple exclusion process in a critical scaling parametrized by a≥0, which creates a shock in the particle density of order aT−1/3, T the observation time. When starting from step initial data, we provide bounds on the limiting law which in particular imply that in the double limit lima→∞limT→∞ one recovers the product limit law and the degeneration of the correlation length observed at shocks of order 1. This result is shown to apply to a general last-passage percolation model. We also obtain bounds on the two-point functions of several airy processes.
AU - Nejjar, Peter
ID - 70
IS - 2
JF - Latin American Journal of Probability and Mathematical Statistics
SN - 1980-0436
TI - Transition to shocks in TASEP and decoupling of last passage times
VL - 15
ER -
TY - JOUR
AB - We consider last passage percolation (LPP) models with exponentially distributed random variables, which are linked to the totally asymmetric simple exclusion process (TASEP). The competition interface for LPP was introduced and studied in Ferrari and Pimentel (2005a) for cases where the corresponding exclusion process had a rarefaction fan. Here we consider situations with a shock and determine the law of the fluctuations of the competition interface around its deter- ministic law of large number position. We also study the multipoint distribution of the LPP around the shock, extending our one-point result of Ferrari and Nejjar (2015).
AU - Ferrari, Patrik
AU - Nejjar, Peter
ID - 447
JF - Revista Latino-Americana de Probabilidade e Estatística
TI - Fluctuations of the competition interface in presence of shocks
VL - 9
ER -