First-order transition for the (1+1)-dimensional q⩾4 Potts model from finite lattice extrapolation

1983 ◽  
Vol 16 (15) ◽  
pp. 2833-2845 ◽  
Author(s):  
F Igloi ◽  
J Solyom
Keyword(s):  
Potts Model ◽  
Finite Lattice ◽  
Order Transition ◽  
First Order ◽  
First Order Transition

NUMERICAL EVIDENCE OF A FIRST-ORDER TRANSITION IN A BRANCHED POLYMER: FINITE SIZE EFFECT AT THE TRANSITION

1992 ◽  
Vol 06 (08) ◽  
pp. 1193-1207 ◽  
Author(s):  
P.D. GUJRATI ◽  
Y. ZHU
Keyword(s):  
Thermodynamic Limit ◽  
Finite Lattice ◽  
Finite Size ◽  
Order Transition ◽  
Linear Polymer ◽  
Finite Size Effect ◽  
Continuous Transition ◽  
Branched Polymer ◽  
First Order ◽  
First Order Transition

Within the ε-expansion scheme, we argue that the upper critical dimension for the continuous transition (if it exists) in the dilute limit of branched polymers should be du=4. However, Monte-Carlo simulations in d=2, 3 and 4 for a single branched polymer on a finite lattice show evidence of a first-order transition at K=K0 and not a continuous transition which we observe for a linear polymer. Here, K represents bond activity. Our extensive simulation in d=2 is compared with previously known results. The existence of a first-order transition at K0 is not inconsistent with power-law singularities as [Formula: see text], which have been observed by various workers. However, for K<K0, the self-similarity which is a prerequisite of a fractal (i.e., a critical) object is only approximate. At K=K0 the branched polymer is a regular, compact object and not a fractal. In constrast, a linear polymer is a fractal at the transition, as is expected and as our simulation also suggests. Thus, our work shows that the order of the thermodynamic limit N→∞ and K→K0 is important, provided the transition remains first-order in the thermodynamic limit. In previous works, one considers [Formula: see text] after the limit N→∞ has already been taken, whereas we are interested in K→K0, followed by N→∞.

scholarly journals POTTS MODEL WITH INVISIBLE COLORS: RANDOM-CLUSTER REPRESENTATION AND PIROGOV–SINAI ANALYSIS

2012 ◽  
Vol 24 (02) ◽  
pp. 1250004 ◽  
Author(s):  
AERNOUT C. D. VAN ENTER ◽  
GIULIO IACOBELLI ◽  
SIAMAK TAATI
Keyword(s):  
Low Temperature ◽  
Symmetry Breaking ◽  
Potts Model ◽  
Temperature Regime ◽  
Order Transition ◽  
Ferromagnetic Interaction ◽  
Random Cluster ◽  
First Order ◽  
First Order Transition ◽  
Cluster Representation

We study a variant of the ferromagnetic Potts model, recently introduced by Tamura, Tanaka and Kawashima, consisting of a ferromagnetic interaction among q "visible" colors along with the presence of r non-interacting "invisible" colors. We introduce a random-cluster representation for the model, for which we prove the existence of a first-order transition for any q > 0, as long as r is large enough. When q > 1, the low-temperature regime displays a q-fold symmetry breaking. The proof involves a Pirogov–Sinai analysis applied to this random-cluster representation of the model.

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