Dilute O($n$) Model & Potts Model: Critical Properties

[jal]

Has anyone here worked with the Potts model so that they can explain why these two papers, which seem so unrelated can use the same model.
http://arxiv.org/abs/0805.2678
Critical properties of a dilute O($n$) model on the kagome lattice
Authors: Biao Li, Wenan Guo, Henk W.J. Blöte
(Submitted on 17 May 2008)
A critical dilute O($n$) model on the kagome lattice is investigated analytically and numerically. We employ a number of exact equivalences which, in a few steps, link the critical O($n$) spin model on the kagome lattice to the exactly solvable critical $q$-state Potts model on the honeycomb lattice with $q=(n+1)^2$.
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http://arxiv.org/abs/0806.3506
Shaken, but not stirred – Potts model coupled to quantum gravity
Authors: J. Ambjorn, K.N. Anagnostopoulos, R. Loll, I. Pushkina
(Submitted on 21 Jun 2008)
We investigate the critical behaviour of both matter and geometry of the three-state Potts model coupled to two-dimensional Lorentzian quantum gravity in the framework of causal dynamical triangulations
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[jal]

I think that I’ve found the answer to my question and more.
I also, got a better understanding of what Garrett could be trying to do with E8.
Geee! … I might be helping everyone with their learning curve.

So … The Kagome lattice finds a relationship to the Potts model …
…. casual dynamic triangulations and quantum gravity finds a relationship with Potts model because of Virasoro Algebra

…. So …now … I watch and learn from the “math kid” …. They should be able to see the relationships that I cannot.
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http://en.wikipedia.org/wiki/Virasoro_algebra
m = 5: c = 4/5. There are 10 representations, which are related to the 3-state Potts model.
m = 6: c = 6/7. There are 15 representations, which are related to the tri critical 3-state Potts model.
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http://www.emis.de/journals/SIGMA/2007/008/
The Virasoro Algebra and Some Exceptional Lie and Finite Groups
Michael P. Tuite
Department of Mathematical Physics, National University of Ireland, Galway, Ireland
Received October 09, 2006, in final form December 16, 2006; Published online January 08, 2007
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Way back when Baez had the following comment:
http://math.ucr.edu/home/baez/week124.html
October 23, 1998
• the Virasoro algebra (which is closely related to the Lie algebra of the group of conformal transformations, and has a representation on the Hilbert space of states of any conformal field theory),
• minimal models (roughly, conformal field theories whose Hilbert space is built from finitely many irreducible representations of the Virasoro algebra),
Richard Block was also the first to write anything about what's now called the Virasoro algebra — a Lie algebra that plays a key role in string theory.
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http://arxiv.org/abs/0709.2539
Multi-dimensional Virasoro algebra and quantum gravity
Authors: T. A. Larsson
(Submitted on 17 Sep 2007)
I review the multi-dimensional generalizations of the Virasoro algebra, i.e. the non-central Lie algebra extensions of the algebra vect(N) of general vector fields in N dimensions, and its Fock representations. Being the Noether symmetry of background independent theories such as N-dimensional general relativity, this algebra is expected to be relevant to the quantization of gravity. To this end, more complicated modules which depend on dynamics in the form of Euler-Lagrange equations are described. These modules can apparently only be interpreted as quantum fields if spacetime has four dimensions and both bosons and fermions are present.
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[Entropia]

The dilute O($n$) model and Potts model are two mathematical models used to study critical phenomena, such as phase transitions, in statistical mechanics. Both models have been extensively studied and have been found to exhibit similar critical properties. This can be explained by the fact that the two models are actually closely related to each other.

The paper by Li, Guo, and Blöte (2008) investigates the critical properties of a dilute O($n$) model on the kagome lattice. They use a number of exact equivalences to connect the critical O($n$) spin model on the kagome lattice to the exactly solvable critical $q$-state Potts model on the honeycomb lattice. This means that the two models are equivalent at their critical points, and thus they exhibit the same critical properties.

Similarly, the paper by Ambjorn et al. (2008) also explores the critical behavior of the Potts model, but in a different context. They couple the Potts model to two-dimensional Lorentzian quantum gravity and study the critical behavior of both matter and geometry. They find that the critical behavior of the Potts model is in agreement with the critical behavior of the dilute O($n$) model, further supporting the idea that these two models are closely related.

In summary, the dilute O($n$) model and Potts model are two different models that can be connected through exact equivalences. This means that they exhibit the same critical properties, despite seemingly unrelated contexts. This highlights the power and versatility of these models in studying critical phenomena in statistical mechanics.