scholarly journals A note on recognizing an old friend in a new place: list coloring and the zero-temperature Potts model

2014 ◽  
Vol 1 (4) ◽  
pp. 429-442
Author(s):  
Joanna Ellis-Monaghan ◽  
Iain Moffatt
Keyword(s):  
Potts Model ◽  
List Coloring ◽  
Zero Temperature

scholarly journals PERSISTENCE IN THE ZERO-TEMPERATURE DYNAMICS OF THE Q-STATES POTTS MODEL ON UNDIRECTED-DIRECTED BARABÁSI–ALBERT NETWORKS AND ERDÖS–RÉNYI RANDOM GRAPHS

2008 ◽  
Vol 19 (12) ◽  
pp. 1777-1785 ◽  
Author(s):  
F. P. FERNANDES ◽  
F. W. S. LIMA
Keyword(s):  
Random Graphs ◽  
Potts Model ◽  
Finite Size ◽  
Glauber Dynamics ◽  
The Other ◽  
Finite Size Effect ◽  
Zero Temperature ◽  
Temperature Dynamics ◽  
Short Times ◽  
Q Values

The zero-temperature Glauber dynamics is used to investigate the persistence probability P(t) in the Potts model with Q = 3, 4, 5, 7, 9, 12, 24, 64, 128, 256, 512, 1024, 4096, 16 384, …, 230 states on directed and undirected Barabási–Albert networks and Erdös–Rényi (ER) random graphs. In this model, it is found that P(t) decays exponentially to zero in short times for directed and undirected ER random graphs. For directed and undirected BA networks, in contrast it decays exponentially to a constant value for long times, i.e., P(∞) is different from zero for all Q values (here studied) from Q = 3, 4, 5, …, 230; this shows "blocking" for all these Q values. Except that for Q = 230 in the undirected case P(t) tends exponentially to zero; this could be just a finite-size effect since in the other "blocking" cases you may have only a few unchanged spins.

scholarly journals Counting Proper Colourings in 4-Regular Graphs via the Potts Model

10.37236/7743 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Ewan Davies
Keyword(s):  
Partition Function ◽  
Internal Energy ◽  
Lower Bounds ◽  
Regular Graph ◽  
Potts Model ◽  
Temperature Limit ◽  
Upper And Lower Bounds ◽  
Regular Graphs ◽  
Zero Temperature ◽  
First Case

We give tight upper and lower bounds on the internal energy per particle in the antiferromagnetic $q$-state Potts model on $4$-regular graphs, for $q\ge 5$. This proves the first case of a conjecture of the author, Perkins, Jenssen and Roberts, and implies tight bounds on the antiferromagnetic Potts partition function. The zero-temperature limit gives upper and lower bounds on the number of proper $q$-colourings of $4$-regular graphs, which almost proves the case $d=4$ of a conjecture of Galvin and Tetali. For any $q \ge 5$ we prove that the number of proper $q$-colourings of a $4$-regular graph is maximised by a union of $K_{4,4}$'s.  

ZERO-TEMPERATURE SKEWED CHIRAL POTTS MODEL

1994 ◽  
Vol 06 (05a) ◽  
pp. 869-885
Author(s):  
R. J. BAXTER
Keyword(s):  
Potts Model ◽  
Shift Operator ◽  
Temperature Limit ◽  
Zero Temperature ◽  
Chiral Potts Model ◽  
Functional Relations ◽  
Spurious Solutions ◽  
Intermediate Phases ◽  
Free Interfaces ◽  
Ice Model

Functional relations have previously been obtained for the eigenvalues of the transfer matrices of the chiral Potts model. Introducing skewed boundary conditions is equivalent to merely modifying the quantum number of the spin shift operator in the relations (which accounts for at least some of the previously noted "spurious solutions"). As a first step towards calculating the general interfacial tension, we consider the model in a zero-temperature limit. It is still non-trivial, there being near-vertical free interfaces separating domains of different spin value. These interfaces behave like the "lines of down arrows" in the ice model, so one may hope to follow Lieb and use the Bethe ansatz to evaluate the partition function. It turns out that this can indeed be done. There is no wetting of an interface by intermediate phases.

FINITE TEMPERATURE ENTROPY OF THE ANTIFERROMAGNETIC POTTS MODEL

1988 ◽  
Vol 02 (06) ◽  
pp. 1495-1501 ◽  
Author(s):  
X. Y. CHEN ◽  
C. Y. PAN
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Potts Model ◽  
Finite Temperature ◽  
Three Dimensions ◽  
Universal Relation ◽  
Zero Temperature ◽  
Disorder Transition ◽  
Color Problem ◽  
Antiferromagnetic Potts Model

Monte Carlo simulation is used to deal with the finite temperature entropy of the q-state antiferromagnetic Potts model which is the extension of the general q-color problem (at zero temperature). The finite temperature entropy of the model in two and three dimensions is obtained which is consistent with the zero temperature results. A possible universal relation of the model to determine when the order-disorder transition happens is proposed.

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