Patrik L. Ferrari and Peter Nejjar Shock fluctuations in flat TASEP under critical scaling J. Stat. Phys., 160(4): 985-1004 2015 http://arxiv.org/abs/1408.4850
Abstract: We consider TASEP with two types of particles starting at every second site. Particles to the left of the origin have jump rate $1$, while particles to the right have jump rate $\alpha$. When $\alpha<1$ there is a formation of a shock where the density jumps to $(1-\alpha)/2$. For $\alpha<1$ fixed, the statistics of the associated height functions around the shock is asymptotically (as time $t\to\infty$) a maximum of two independent random variables as shown in [arXiv:1306.3336]. In this paper we consider the critical scaling when $1-\alpha=a t^{-1/3}$, where $t\gg 1$ is the observation time. In that case the decoupling does not occur anymore. We determine the limiting distributions of the shock and numerically study its convergence as a function of $a$. We see that the convergence to product $F_{\rm GOE}^2$ occurs quite rapidly as $a$ increases. The critical scaling is analogue to the one used in the last passage percolation to obtain the BBP transition processes.
Herbert Koch and Nikolai Nadirashvili Partial analyticity and nodal sets for nonlinear elliptic systems 2015 http://arxiv.org/abs/1506.06224
Barbara Niethammer, Sebastian Throm and Juan J. L. Velázquez A revised proof of uniqueness of self-similar profiles to Smoluchowski's coagulation equation for kernels close to constant 2015 http://arxiv.org/abs/1510.03361
Barbara Niethammer, Juan J. L. Velázquez and Michael Helmers Mathematical analysis of a coarsening model with local interactions 2015 http://arxiv.org/abs/1509.04917
Barbara Niethammer, Sebastian Throm and Juan J. L. Velázquez Self-similar solutions with fat tails for Smoluchowski's coagulation equation with singular kernels Ann. I. H. Poincaré - AN 2015 http://dx.doi.org/10.1016/j.anihpc.2015.04.002
2014
Thomas Hangelbroek, Francis J. Narcowich, Christian Rieger and Joseph D. Ward An inverse theorem on bounded domains for meshless methods using localized bases 2014 http://arxiv.org/pdf/1406.1435v1
Barbara Niethammer and Juan J. L. Velázquez Exponential tail behaviour of self-similar solutions to Smoluchowski's coagulation equation Communications in Partial Differential Equations, 39(12): 2314-2350 2014 http://dx.doi.org/10.1080/03605302.2014.918143
Abstract: We consider self-similar solutions with finite mass to Smoluchowski's coagulation equation for rate kernels that have homogeneity zero but are possibly singular such as Smoluchowski's original kernel. We prove pointwise exponential decay of these solutions under rather mild assumptions on the kernel. If the support of the kernel is sufficiently large around the diagonal we also proof that \( \lim_{x\rightarrow\infty}\frac{1}{x}\log\left(\frac{1}{f(x)}\right)\) exists. In addition we prove properties of the prefactor if the kernel is uniformly bounded below.
Barbara Niethammer and Juan J. L. Velázquez Uniqueness of self-similar solutions to Smoluchowski's coagulation equations for kernels that are close to constant J. Stat. Phys., 157(1): 158-181 2014 http://dx.doi.org/10.1007/s10955-014-1070-3
Abstract: We consider self-similar solutions to Smoluchowski's coagulation equation for kernels K=K(x,y) that are homogeneous of degree zero and close to constant in the sense that \[ -\varepsilon \leq K(x,y)-2 \leq \varepsilon \Big(\Big(\frac{x}{y}\Big)^{\alpha} + \Big(\frac{y}{x}\Big)^{\alpha}\Big) \] for \(\alpha \in [0,1)\). We prove that self-similar solutions with given mass are unique if \(\varepsilon\) is sufficiently small which is the first such uniqueness result for kernels that are not solvable. Our proof relies on a contraction argument in a norm that measures the distance of solutions with respect to the weak topology of measures.
0
P.L. Ferrari and P. Nejjar The second class particle process at shocks preprint: arXiv:2309.09570 0 https://arxiv.org/abs/2309.09570
Abstract: We consider the totally asymmetric simple exclusion process (TASEP) starting with a shock discontinuity at the origin, with asymptotic densities λ to the left of the origin and Ï to the right of it and λ<Ï. We find an exact identity for the distribution of a second class particle starting at the origin. Then we determine the limiting joint distributions of the second class particle. Bypassing the last passage percolation model, we work directly in TASEP, allowing us to extend previous one-point distribution results via a more direct and shorter ansatz.
P. Nejjar KPZ statistics of second class particles in ASEP via mixing preprint:arXiv:1911.09426 0 https://arxiv.org/abs/1911.09426
Abstract: We consider the asymmetric simple exclusion process on Z with a single second class particle initially at the origin. The first class particles form two rarefaction fans which come together at the origin, where the large time density jumps from 0 to 1. We are interested in X(t), the position of the second class particle at time t. We show that, under the KPZ 1/3 scaling, X(t) is asymptotically distributed as the difference of two independent, GUE-distributed random variables.