Timeline for For which values of $N$ is known the Lieb-Simon Inequality for $Z_N$ Models ?
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 10, 2010 at 22:18 | comment | added | Leandro | Thank you Velenik for all the informations. After your previous post, I finished to check that Simon's approach can not be used to prove the inequality (even in dimension 2). Unfortunatelly we can not conclude that the inequality is false. Similarly, I saw that the Simon's technique also does not work for any odd $N$. But I went nowhere for even values of $N$. | |
| Jun 10, 2010 at 7:09 | comment | added | Yvan Velenik | Actually, the general $N$ case would rather be a clock model, and I don't know what is the order of the phase transition for this model (of course, for $q=3$ the clock and Potts model coincide, so my previous comment still indicate that such a result should not be true in dimensions $d\geq 3$ in this case). | |
| Jun 10, 2010 at 6:57 | comment | added | Yvan Velenik | Actually, the strong form cannot hold for general $q$-states Potts models. Indeed, it would imply (see the paper by Lieb) that the massgap is continuous at the phase transition, a fact that is known not to hold for large values of $q$. Of course, one might still believe that (i) the strong form holds for small values of $q$ (which should be $q\leq 4$ in dimension $2$, but only $q=2$, i.e. the Ising model, when $d\geq 3$), or (ii) the weak form holds more generally... | |
| Jun 9, 2010 at 19:46 | comment | added | Leandro | Yes, I am doing that. But in this case it leads more challenge comparison between the multigraphs and I am not able to do that. For this $Z_N$ model we have to take in account oriented multigraphs and also the Euler circuits that rises in the simon proof now are more general because it is allowed the degree to be zero modulus $N$. If you remember the reference, please let me know. I will think about to post the multigraph problem here also. Thanks | |
| Jun 9, 2010 at 18:43 | comment | added | Yvan Velenik | Right. I misread your question, sorry. I'll think about it (no ref come immediately to mind, though). Would you settle for the weak form (infinite volume measure and possibly some ($\beta$ and $N$ dependent) prefactor?). It is pretty clear how to derive such a result for $\beta < \beta_c$ in Potts model, but I have no idea how to treat general values of $\beta$ (the graphical representations available don't seem nice enough, but I should think more). On the other hand, I would simply look at the proofs of Lieb, Rivasseau, etc., to see whether they can be extended to that case... | |
| Jun 9, 2010 at 15:24 | comment | added | Leandro | Because in $N=3$ this model should be equivalent to 3 state Potts model. But I never locate any reference proving this statement. | |
| Jun 9, 2010 at 14:08 | comment | added | Leandro | Hi Velenik, perhaps you thought about $N$ as the dimension where spin variable lives and in this case I agree with you about the results. But here I am using $N$ to denote the number of states. I miss to add that $N=2$ here is the Ising model, where the inequality holds and I also Know that is valid for the XY model as you mention (Lieb+Rivasseua) that would be something like the limit when $N$ goes to infinity. Because this last observation I was think about that could be reasonable to prove it for large $N$ at least. I also heard one time that it could not be valid for $N=3$. | |
| Jun 9, 2010 at 13:58 | vote | accept | Leandro | ||
| Jun 9, 2010 at 14:00 | |||||
| Jun 9, 2010 at 8:21 | history | answered | Yvan Velenik | CC BY-SA 2.5 |