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Open AccessFeature PaperArticle

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

by Nickolay Izmailian 1,2,* and Ralph Kenna 3,†
1
A. Alikhanyan National Laboratory (Yerevan Physics Institute), Alikhanian Brothers 2, Yerevan 375036, Armenia
2
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia
3
Statistical Physics Group, Centre for Fluid and Complex Systems, Coventry University, Coventry CV1 5FB, UK
*
Author to whom correspondence should be addressed.
(Dated: 10 September 2019).
Entropy 2019, 21(9), 895; https://doi.org/10.3390/e21090895
Received: 14 August 2019 / Revised: 10 September 2019 / Accepted: 11 September 2019 / Published: 15 September 2019
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 c = 2 ) 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. View Full-Text
Keywords: universality; corrections to scaling; corners; spanning tree; cobwebm fan; Ivashkevich-Izmailian-Hu algorithm universality; corrections to scaling; corners; spanning tree; cobwebm fan; Ivashkevich-Izmailian-Hu algorithm
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Izmailian, N.; Kenna, R. Universality and Exact Finite-Size Corrections for Spanning Trees on Cobweb and Fan Networks. Entropy 2019, 21, 895.

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