{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:31:38Z","timestamp":1760149898739,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,9,9]],"date-time":"2023-09-09T00:00:00Z","timestamp":1694217600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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 (\u201croot\u201d) 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\u22122)\/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\u22121N 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.<\/jats:p>","DOI":"10.3390\/e25091318","type":"journal-article","created":{"date-parts":[[2023,9,11]],"date-time":"2023-09-11T08:58:08Z","timestamp":1694422688000},"page":"1318","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Path Counting on Tree-like Graphs with a Single Entropic Trap: Critical Behavior and Finite Size Effects"],"prefix":"10.3390","volume":"25","author":[{"given":"Alexey V.","family":"Gulyaev","sequence":"first","affiliation":[{"name":"Independent Researcher, 119234 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3168-1307","authenticated-orcid":false,"given":"Mikhail V.","family":"Tamm","sequence":"additional","affiliation":[{"name":"CUDAN Open Lab and School of Digital Technologies, Tallinn University, 10120 Tallinn, Estonia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Newman, M.E.J. 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