| Autor: |
Alexey V. Gulyaev, Mikhail V. Tamm |
| Jazyk: |
angličtina |
| Rok vydání: |
2023 |
| Předmět: |
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| Zdroj: |
Entropy, Vol 25, Iss 9, p 1318 (2023) |
| Druh dokumentu: |
article |
| ISSN: |
1099-4300 |
| DOI: |
10.3390/e25091318 |
| Popis: |
It is known that maximal entropy random walks and partition functions that count long paths on graphs tend to become localized near nodes with a high degree. Here, we revisit the simplest toy model of such a localization: a regular tree of degree p with one special node (“root”) that has a degree different from all the others. We present an in-depth study of the path-counting problem precisely at the localization transition. We study paths that start from the root in both infinite trees and finite, locally tree-like regular random graphs (RRGs). For the infinite tree, we prove that the probability distribution function of the endpoints of the path is a step function. The position of the step moves away from the root at a constant velocity v=(p−2)/p. We find the width and asymptotic shape of the distribution in the vicinity of the shock. For a finite RRG, we show that a critical slowdown takes place, and the trajectory length needed to reach the equilibrium distribution is on the order of N instead of logp−1N away from the transition. We calculate the exact values of the equilibrium distribution and relaxation length, as well as the shapes of slowly relaxing modes. |
| Databáze: |
Directory of Open Access Journals |
| Externí odkaz: |
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