Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe Limiting Behavior of the FTASEP with Product Bernoulli Initial Distribution
72
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study the facilitated totally asymmetric exclusion process on the one dimensional integer lattice. We investigate the invariant measures and the limiting behavior of the process. We mainly derive the limiting distribution of the process when the initial distribution is the Bernoulli product measure with density $1/2$. We also prove that in the low density regime, the system finally converges to an absorbing state.
rate research
Read More
Place an obstacle with probability $1-p$ independently at each vertex of $mathbb Z^d$ and consider a simple symmetric random walk that is killed upon hitting one of the obstacles. For $d geq 2$ and $p$ strictly above the critical threshold for site percolation, we condition on the environment such that the origin is contained in an infinite connected component free of obstacles. It has previously been shown that with high probability, the random walk conditioned on survival up to time $n$ will be localized in a ball of volume asymptotically $dlog_{1/p}n$. In this work, we prove that this ball is free of obstacles, and we derive the limiting one-time distributions of the random walk conditioned on survival. Our proof is based on obstacle modifications and estimates on how such modifications affect the probability of the obstacle configurations as well as their associated Dirichlet eigenvalues, which is of independent interest.
The relaxation time limit of the one-point distribution of the spatially periodic totally asymmetric simple exclusion process is expected to be the universal one point distribution for the models in the KPZ universality class in a periodic domain. Unlike the infinite line case, the limiting one point distribution depends non-trivially on the scaled time parameter. We study several properties of this distribution for the case of the periodic step and flat initial conditions. We show that the distribution changes from a Tracy-Widom distribution in the small time limit to the Gaussian distribution in the large time limit, and also obtain right tail estimate for all time. Furthermore, we establish a connection to integrable differential equations such as the KP equation, coupled systems of mKdV and nonlinear heat equations, and the KdV equation.
Consider two high-dimensional random vectors $widetilde{mathbf x}inmathbb R^p$ and $widetilde{mathbf y}inmathbb R^q$ with finite rank correlations. More precisely, suppose that $widetilde{mathbf x}=mathbf x+Amathbf z$ and $widetilde{mathbf y}=mathbf y+Bmathbf z$, for independent random vectors $mathbf xinmathbb R^p$, $mathbf yinmathbb R^q$ and $mathbf zinmathbb R^r$ with iid entries of mean 0 and variance 1, and two deterministic matrices $Ainmathbb R^{ptimes r}$ and $Binmathbb R^{qtimes r}$ . With $n$ iid observations of $(widetilde{mathbf x},widetilde{mathbf y})$, we study the sample canonical correlations between them. In this paper, we focus on the high-dimensional setting with a rank-$r$ correlation. Let $t_1gecdotsge t_r$ be the squares of the population canonical correlation coefficients (CCC) between $widetilde{mathbf x}$ and $widetilde{mathbf y}$, and $widetildelambda_1gecdotsgewidetildelambda_r$ be the squares of the largest $r$ sample CCC. Under certain moment assumptions on the entries of $mathbf x$, $mathbf y$ and $mathbf z$, we show that there exists a threshold $t_cin(0, 1)$ such that if $t_i>t_c$, then $sqrt{n}(widetildelambda_i-theta_i)$ converges in law to a centered normal distribution, where $theta_i>lambda_+$ is a fixed outlier location determined by $t_i$. Our results extend the ones in [4] for Gaussian vectors. Moreover, we find that the variance of the limiting distribution of $sqrt{n}(widetildelambda_i-theta_i)$ also depends on the fourth cumulants of the entries of $mathbf x$, $mathbf y$ and $mathbf z$, a phenomenon that cannot be observed in the Gaussian case.
We substantially refine asymptotic logarithmic upper bounds produced by Svante Janson (2015) on the right tail of the limiting QuickSort distribution function $F$ and by Fill and Hung (2018) on the right tails of the corresponding density $f$ and of the absolute derivatives of $f$ of each order. For example, we establish an upper bound on $log[1 - F(x)]$ that matches conjectured asymptotics of Knessl and Szpankowski (1999) through terms of order $(log x)^2$; the corresponding order for the Janson (2015) bound is the lead order, $x log x$. Using the refined asymptotic bounds on $F$, we derive right-tail large deviation (LD) results for the distribution of the number of comparisons required by QuickSort that substantially sharpen the two-sided LD results of McDiarmid and Hayward (1996).
In a continuous-time setting, Fill (2010) proved, for a large class of probabilistic sources, that the number of symbol comparisons used by QuickSort, when centered by subtracting the mean and scaled by dividing by time, has a limiting distribution, but proved little about that limiting random variable Y -- not even that it is nondegenerate. We establish the nondegeneracy of Y. The proof is perhaps surprisingly difficult.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
