{"title":"Critical Dynamics of Random Surfaces","authors":"Christof Schmidhuber","doi":"arxiv-2409.05547","DOIUrl":null,"url":null,"abstract":"Conformal field theories with central charge $c\\le1$ on random surfaces have\nbeen extensively studied in the past. Here, this discussion is extended from\ntheir equilibrium distribution to their critical dynamics. This is motivated by\nthe conjecture that these models describe the time evolution of certain social\nnetworks that are self-driven to a critical point. The time evolution of the\nsurface area is identified as a Cox Ingersol Ross process. Planar surfaces\nshrink, while higher genus surfaces grow until the cosmological constant stops\ntheir growth. Three different equilibrium states are distingushed, dominated by\n(i) small planar surfaces, (ii) large surfaces with high but finite genus, and\n(iii) foamy surfaces, whose genus diverges. Time variations of the order\nparameter are analyzed and are found to have generalized hyperbolic\ndistributions. In state (i), those have power law tails with a tail index close\nto 4. Analogies between the time evolution of the order parameter and a\nmultifractal random walk are also pointed out.","PeriodicalId":501139,"journal":{"name":"arXiv - QuantFin - Statistical Finance","volume":"20 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - QuantFin - Statistical Finance","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.05547","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Conformal field theories with central charge $c\le1$ on random surfaces have
been extensively studied in the past. Here, this discussion is extended from
their equilibrium distribution to their critical dynamics. This is motivated by
the conjecture that these models describe the time evolution of certain social
networks that are self-driven to a critical point. The time evolution of the
surface area is identified as a Cox Ingersol Ross process. Planar surfaces
shrink, while higher genus surfaces grow until the cosmological constant stops
their growth. Three different equilibrium states are distingushed, dominated by
(i) small planar surfaces, (ii) large surfaces with high but finite genus, and
(iii) foamy surfaces, whose genus diverges. Time variations of the order
parameter are analyzed and are found to have generalized hyperbolic
distributions. In state (i), those have power law tails with a tail index close
to 4. Analogies between the time evolution of the order parameter and a
multifractal random walk are also pointed out.