scholarly journals Non-existence of bi-infinite geodesics in the exponential corner growth model

2020 ◽  
Vol 8 ◽  
Author(s):  
Márton Balázs ◽  
Ofer Busani ◽  
Timo Seppäläinen
Keyword(s):  
Growth Model ◽  
Coarse Graining ◽  
Exponential Weights ◽  
Percolation Process ◽  
Last Passage Percolation ◽  
Corner Growth Model ◽  
And Control

Abstract This paper gives a self-contained proof of the non-existence of nontrivial bi-infinite geodesics in directed planar last-passage percolation with exponential weights. The techniques used are couplings, coarse graining, and control of geodesics through planarity and estimates derived from increment-stationary versions of the last-passage percolation process.

Percolation phase transition of static and growing networks under a weighted function

2016 ◽  
Vol 27 (07) ◽  
pp. 1650082 ◽  
Author(s):  
Xiao Jia ◽  
Jin-Song Hong ◽  
Ya-Chun Gao ◽  
Hong-Chun Yang ◽  
Chun Yang ◽  
...  
Keyword(s):  
Phase Transition ◽  
Continuous Phase ◽  
Weighted Function ◽  
Percolation Process ◽  
Practical Applications ◽  
Network Intervention ◽  
Growing Networks ◽  
Discontinuous Phase ◽  
And Control ◽  
Growing Network

We investigate the percolation phase transitions in both the static and growing networks where the nodes are sampled according to a weighted function with a tunable parameter [Formula: see text]. For the static network, i.e. the number of nodes is constant during the percolation process, the percolation phase transition can evolve from continuous to discontinuous as the value of [Formula: see text] is tuned. Based on the properties of the weighted function, three typical values of [Formula: see text] are analyzed. The model becomes the classical Erdös–Rényi (ER) network model at [Formula: see text]. When [Formula: see text], it is shown that the percolation process generates a weakly discontinuous phase transition where the order parameter exhibits an extremely abrupt transition with a significant jump in large but finite system. For [Formula: see text], the cluster size distribution at the lower pseudo-transition point does not obey the power-law behavior, indicating a strongly discontinuous phase transition. In the case of growing network, in which the collection of nodes is increasing, a smoother continuous phase transition emerges at [Formula: see text], in contrast to the weakly discontinuous phase transition of the static network. At [Formula: see text], on the other hand, probability modulation effect shows that the nature of strongly discontinuous phase transition remains the same with the static network despite the node arrival even in the thermodynamic limit. These percolation properties of the growing networks could provide useful reference for network intervention and control in practical applications in consideration of the increasing size of most actual networks.

scholarly journals Non-existence of bi-infinite geodesics in the exponential corner growth model – Corrigendum

2021 ◽  
Vol 9 ◽  
Author(s):  
Márton Balázs ◽  
Ofer Busani ◽  
Timo Seppäläinen
Keyword(s):  
Growth Model ◽  
Corner Growth Model

scholarly journals Flats, spikes and crevices: the evolving shape of the inhomogeneous corner growth model

2021 ◽  
Vol 26 (none) ◽  
Author(s):  
Elnur Emrah ◽  
Christopher Janjigian ◽  
Timo Seppäläinen
Keyword(s):  
Growth Model ◽  
Corner Growth Model

scholarly journals Stationary cocycles and Busemann functions for the corner growth model

2016 ◽  
Vol 169 (1-2) ◽  
pp. 177-222 ◽  
Author(s):  
Nicos Georgiou ◽  
Firas Rassoul-Agha ◽  
Timo Seppäläinen
Keyword(s):  
Growth Model ◽  
Busemann Functions ◽  
Corner Growth Model

scholarly journals Cube Root Fluctuations for the Corner Growth Model Associated to the Exclusion Process

2006 ◽  
Vol 11 (0) ◽  
pp. 1094-1132 ◽  
Author(s):  
Marton Balazs ◽  
Eric Cator ◽  
Timo Seppalainen
Keyword(s):  
Growth Model ◽  
Exclusion Process ◽  
Cube Root ◽  
Corner Growth Model
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