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Universality and Exact Finite-Size Corrections for Spanning Trees on Cobweb and Fan Networks

The concept of universality is a cornerstone of theories of critical phenomena. It is very well understood in most systems, especially in the thermodynamic limit. Finite-size systems present additional challenges. Even in low dimensions, universality of the edge and corner contributions to free ener...

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Autores principales: Izmailian, Nickolay, Kenna, Ralph
Formato: Online Artículo Texto
Lenguaje:English
Publicado: MDPI 2019
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7515416/
http://dx.doi.org/10.3390/e21090895
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author Izmailian, Nickolay
Kenna, Ralph
author_facet Izmailian, Nickolay
Kenna, Ralph
author_sort Izmailian, Nickolay
collection PubMed
description The concept of universality is a cornerstone of theories of critical phenomena. It is very well understood in most systems, especially in the thermodynamic limit. Finite-size systems present additional challenges. Even in low dimensions, universality of the edge and corner contributions to free energies and response functions is less investigated and less well understood. In particular, the question arises of how universality is maintained in correction-to-scaling in systems of the same universality class but with very different corner geometries. Two-dimensional geometries deliver the simplest such examples that can be constructed with and without corners. To investigate how the presence and absence of corners manifest universality, we analyze the spanning tree generating function on two different finite systems, namely the cobweb and fan networks. The corner free energies of these configurations have stimulated significant interest precisely because of expectations regarding their universal properties and we address how this can be delivered given that the finite-size cobweb has no corners while the fan has four. To answer, we appeal to the Ivashkevich–Izmailian–Hu approach which unifies the generating functions of distinct networks in terms of a single partition function with twisted boundary conditions. This unified approach shows that the contributions to the individual corner free energies of the fan network sum to zero so that it precisely matches that of the web. It therefore also matches conformal theory (in which the central charge is found to be [Formula: see text]) and finite-size scaling predictions. Correspondence in each case with results established by alternative means for both networks verifies the soundness of the Ivashkevich–Izmailian–Hu algorithm. Its broad range of usefulness is demonstrated by its application to hitherto unsolved problems—namely the exact asymptotic expansions of the logarithms of the generating functions and the conformal partition functions for fan and cobweb geometries. We also investigate strip geometries, again confirming the predictions of conformal field theory. Thus, the resolution of a universality puzzle demonstrates the power of the algorithm and opens up new applications in the future.
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spelling pubmed-75154162020-11-09 Universality and Exact Finite-Size Corrections for Spanning Trees on Cobweb and Fan Networks Izmailian, Nickolay Kenna, Ralph Entropy (Basel) Article The concept of universality is a cornerstone of theories of critical phenomena. It is very well understood in most systems, especially in the thermodynamic limit. Finite-size systems present additional challenges. Even in low dimensions, universality of the edge and corner contributions to free energies and response functions is less investigated and less well understood. In particular, the question arises of how universality is maintained in correction-to-scaling in systems of the same universality class but with very different corner geometries. Two-dimensional geometries deliver the simplest such examples that can be constructed with and without corners. To investigate how the presence and absence of corners manifest universality, we analyze the spanning tree generating function on two different finite systems, namely the cobweb and fan networks. The corner free energies of these configurations have stimulated significant interest precisely because of expectations regarding their universal properties and we address how this can be delivered given that the finite-size cobweb has no corners while the fan has four. To answer, we appeal to the Ivashkevich–Izmailian–Hu approach which unifies the generating functions of distinct networks in terms of a single partition function with twisted boundary conditions. This unified approach shows that the contributions to the individual corner free energies of the fan network sum to zero so that it precisely matches that of the web. It therefore also matches conformal theory (in which the central charge is found to be [Formula: see text]) and finite-size scaling predictions. Correspondence in each case with results established by alternative means for both networks verifies the soundness of the Ivashkevich–Izmailian–Hu algorithm. Its broad range of usefulness is demonstrated by its application to hitherto unsolved problems—namely the exact asymptotic expansions of the logarithms of the generating functions and the conformal partition functions for fan and cobweb geometries. We also investigate strip geometries, again confirming the predictions of conformal field theory. Thus, the resolution of a universality puzzle demonstrates the power of the algorithm and opens up new applications in the future. MDPI 2019-09-15 /pmc/articles/PMC7515416/ http://dx.doi.org/10.3390/e21090895 Text en © 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
spellingShingle Article
Izmailian, Nickolay
Kenna, Ralph
Universality and Exact Finite-Size Corrections for Spanning Trees on Cobweb and Fan Networks
title Universality and Exact Finite-Size Corrections for Spanning Trees on Cobweb and Fan Networks
title_full Universality and Exact Finite-Size Corrections for Spanning Trees on Cobweb and Fan Networks
title_fullStr Universality and Exact Finite-Size Corrections for Spanning Trees on Cobweb and Fan Networks
title_full_unstemmed Universality and Exact Finite-Size Corrections for Spanning Trees on Cobweb and Fan Networks
title_short Universality and Exact Finite-Size Corrections for Spanning Trees on Cobweb and Fan Networks
title_sort universality and exact finite-size corrections for spanning trees on cobweb and fan networks
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7515416/
http://dx.doi.org/10.3390/e21090895
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