P.L. Ferrari and B. Vető Upper tail decay of KPZ models with Brownian initial conditions preprint,arXiv:2007.13496 2020 https://arxiv.org/abs/2007.13496
Abstract: In this paper we consider the limiting distribution of KPZ growth models with random but not stationary initial conditions introduced in [Chhita-Ferrari-Spohn 2018]. The one-point distribution of the limit is given in terms of a variational problem. By directly studying it, we deduce the right tail asymptotic of the distribution function. This gives a rigorous proof and extends the results obtained in [Meerson-Schmidt 2017].
2019
P.L. Ferrari and B. Vető Fluctuations of the Arctic curve in the tilings of the Aztec diamond on restricted domains preprint: arXiv:1909.10840 2019 https://arxiv.org/abs/1909.10840
Abstract: We consider uniform random domino tilings of the restricted Aztec diamond which is obtained by cutting off an upper triangular part of the Aztec diamond by a horizontal line. The restriction line asymptotically touches the arctic circle that is the limit shape of the north polar region in the unrestricted model. We prove that the rescaled boundary of the north polar region in the restricted domain converges to the Airy$_2$ process conditioned to stay below a parabola with explicit continuous statistics and the finite dimensional distribution kernels. The limit is the hard-edge tacnode process which was first discovered in the framework of non-intersecting Brownian bridges. The proof relies on a random walk representation of the correlation kernel of the non-intersecting line ensemble which corresponds to a random tiling.
2016
P.L. Ferrari and B. Vető The hard-edge tacnode process for Brownian motion preprint, arXiv:1608.00394 2016 https://arxiv.org/abs/1608.00394
Abstract: {We consider N non-intersecting Brownian bridges conditioned to stay below a fixed threshold. We consider a scaling limit where the limit shape is tangential to the threshold. In the large N limit, we determine the limiting distribution of the top Brownian bridge conditioned to stay below a function as well as the limiting correlation kernel of the system. It is a one-parameter family of processes which depends on the tuning of the threshold position on the natural fluctuation scale. We also discuss the relation to the six-vertex model and the Aztec diamond on restricted domains.}
2015
Patrik L. Ferrari and Balint Vető Tracy-Widom asymptotics for q-TASEP Ann. Inst. H. Poincaré Probab. Statist., 51(4): 1465-1485 2015 http://dx.doi.org/10.1214/14-AIHP614
Abstract: We consider the q-TASEP, that is a q-deformation of the totally asymmetric simple exclusion process (TASEP) on \(\mathbb{Z}\) for \(q \in [0,1)\) where the jump rates depend on the gap to the next particle. For step initial condition, we prove that the current fluctuation of \(q\)-TASEP at time \(\tau\) are of order \(\tau^{1/3}\) and asymptotically distributed as the GUE Tracy-Widom distribution.
2014
Alexei Borodin, Ivan Corwin, Patrik L. Ferrari and Balint Vető Height fluctuations for the stationary KPZ equation Math. Phys. Anal. Geom., 18(1, Art. 20): 1-95 2014 http://arxiv.org/abs/1407.6977
Abstract: We compute the one-point probability distribution for the stationary KPZ equation (i.e. initial data $\mathcal{H}(0,X)=B(X)$, for $B(X)$ a two-sided standard Brownian motion) and show that as time $T$ goes to infinity, the fluctuations of the height function $\mathcal{H}(T,X)$ grow like $T^{1/3}$ and converge to those previously encountered in the study of the stationary totally asymmetric simple exclusion process, polynuclear growth model and last passage percolation. The starting point for this work is our derivation of a Fredholm determinant formula for Macdonald processes which degenerates to a corresponding formula for Whittaker processes. We relate this to a polymer model which mixes the semi-discrete and log-gamma random polymers. A special case of this model has a limit to the KPZ equation with initial data given by a two-sided Brownian motion with drift $β$ to the left of the origin and $b$ to the right of the origin. The Fredholm determinant has a limit for $β>b$, and the case where $β=b$ (corresponding to the stationary initial data) follows from an analytic continuation argument.