Finite-size scaling for Potts models |
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Authors: | Christian Borgs Roman Kotecký Salvador Miracle-Solé |
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Institution: | (1) Department of Mathematics, Harvard University, 02138 Cambridge, Massachusetts;(2) Present address: Institut für Theoretische Physik, Freie Universität Berlin, D-1000 Berlin 33, Germany;(3) Department of Mathematical Physics, Charles University, 18000 Prague 8, Czechoslovakia;(4) CNRS, Centre de Physique Théorique, Luminy, Case 907, F-13288 Marseille, France |
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Abstract: | Recently, Borgs and Kotecký developed a rigorous theory of finite-size effects near first-order phase transitions. Here we apply this theory to the ferromagneticq-state Potts model, which (forq large andd2) undergoes a first-order phase transition as the inverse temperature is varied. We prove a formula for the internal energy in a periodic cube of side lengthL which describes the rounding of the infinite-volume jumpE in terms of a hyperbolic tangent, and show that the position of the maximum of the specific heat is shifted by
m
(L)=(Inq/E)L
–d
+O(L
–2d
) with respect to the infinite-volume transition point
t
. We also propose an alternative definition of the finite-volume transition temperature
t
(L) which might be useful for numerical calculations because it differs only by exponentially small corrections from
t
. |
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Keywords: | First-order phase transitions finite-size scaling Potts model Fortuin-Kasteleyn representation |
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