scholarly journals On the Passage Time Geometry of the Last Passage Percolation Problem

Author(s):  
Tom Alberts ◽  
Eric Cator
1993 ◽  
Vol 30 (04) ◽  
pp. 851-862 ◽  
Author(s):  
L. Chayes ◽  
C. Winfield

We introduce and study a novel type of first-passage percolation problem onwhere the associated first-passage time measures the density of interface between two types of sites. If the types, designated + and –, are independently assigned their values with probabilitypand (1 —p) respectively, we show that the density of interface is non-zero provided that both species are subcritical with regard to percolation, i.e.pc>p> 1 –pc.Furthermore, we show that asp↑pcorp↓ (1 –pc), the interface density vanishes with scaling behavior identical to the correlation length of the site percolation problem.


10.37236/1717 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
E. Strahov

A combinatorial interpretation is provided for the moments of characteristic polynomials of random unitary matrices. This leads to a rather unexpected consequence of the Keating and Snaith conjecture: the moments of $|\zeta(1/2+it)|$ turn out to be connected with some increasing subsequence problems (such as the last passage percolation problem).


1993 ◽  
Vol 30 (4) ◽  
pp. 851-862 ◽  
Author(s):  
L. Chayes ◽  
C. Winfield

We introduce and study a novel type of first-passage percolation problem on where the associated first-passage time measures the density of interface between two types of sites. If the types, designated + and –, are independently assigned their values with probability p and (1 — p) respectively, we show that the density of interface is non-zero provided that both species are subcritical with regard to percolation, i.e. pc > p > 1 – pc. Furthermore, we show that as p ↑ pc or p ↓ (1 – pc), the interface density vanishes with scaling behavior identical to the correlation length of the site percolation problem.


1980 ◽  
Vol 45 (3) ◽  
pp. 777-782 ◽  
Author(s):  
Milan Šolc

The establishment of chemical equilibrium in a system with a reversible first order reaction is characterized in terms of the distribution of first passage times for the state of exact chemical equilibrium. The mean first passage time of this state is a linear function of the logarithm of the total number of particles in the system. The equilibrium fluctuations of composition in the system are characterized by the distribution of the recurrence times for the state of exact chemical equilibrium. The mean recurrence time is inversely proportional to the square root of the total number of particles in the system.


Author(s):  
Natalie Packham ◽  
Lutz Schloegl ◽  
Wolfgang M. Schmidt

2011 ◽  
Vol 48 (02) ◽  
pp. 366-388 ◽  
Author(s):  
Eckhard Schlemm

We consider the first passage percolation problem on the random graph with vertex set N x {0, 1}, edges joining vertices at a Euclidean distance equal to unity, and independent exponential edge weights. We provide a central limit theorem for the first passage times l n between the vertices (0, 0) and (n, 0), thus extending earlier results about the almost-sure convergence of l n / n as n → ∞. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to explicitly calculate the asymptotic variance in the central limit theorem.


2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Denis V. Novitsky ◽  
Dmitry Lyakhov ◽  
Dominik Michels ◽  
Dmitrii Redka ◽  
Alexander A. Pavlov ◽  
...  

AbstractUnique and flexible properties of non-Hermitian photonic systems attract ever-increasing attention via delivering a whole bunch of novel optical effects and allowing for efficient tuning light-matter interactions on nano- and microscales. Together with an increasing demand for the fast and spatially compact methods of light governing, this peculiar approach paves a broad avenue to novel optical applications. Here, unifying the approaches of disordered metamaterials and non-Hermitian photonics, we propose a conceptually new and simple architecture driven by disordered loss-gain multilayers and, therefore, providing a powerful tool to control both the passage time and the wave-front shape of incident light with different switching times. For the first time we show the possibility to switch on and off kink formation by changing the level of disorder in the case of adiabatically raising wave fronts. At the same time, we deliver flexible tuning of the output intensity by using the nonlinear effect of loss and gain saturation. Since the disorder strength in our system can be conveniently controlled with the power of the external pump, our approach can be considered as a basis for different active photonic devices.


Author(s):  
Márton Balázs ◽  
Ofer Busani ◽  
Timo Seppäläinen

AbstractWe consider point-to-point last-passage times to every vertex in a neighbourhood of size $$\delta N^{\nicefrac {2}{3}}$$ δ N 2 3 at distance N from the starting point. The increments of the last-passage times in this neighbourhood are shown to be jointly equal to their stationary versions with high probability that depends only on $$\delta $$ δ . Through this result we show that (1) the $$\text {Airy}_2$$ Airy 2 process is locally close to a Brownian motion in total variation; (2) the tree of point-to-point geodesics from every vertex in a box of side length $$\delta N^{\nicefrac {2}{3}}$$ δ N 2 3 going to a point at distance N agrees inside the box with the tree of semi-infinite geodesics going in the same direction; (3) two point-to-point geodesics started at distance $$N^{\nicefrac {2}{3}}$$ N 2 3 from each other, to a point at distance N, will not coalesce close to either endpoint on the scale N. Our main results rely on probabilistic methods only.


Sign in / Sign up

Export Citation Format

Share Document