Timeline for half-plane percolation clusters
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 1, 2014 at 20:52 | answer | added | Achilleas | timeline score: 2 | |
| Feb 18, 2011 at 23:02 | vote | accept | James Propp | ||
| May 4, 2010 at 1:33 | comment | added | James Propp | To answer Tom's question ("What's your motivation?"): I'm interested in David Wilson's "3/8 conjecture" for the Fully Packed Loops model, and my understanding is that, thanks to Cantini and Sportiello's recent proof of the Razumov-Stroganov conjecture, we now know that Wilson's conjecture is equivalent to the conjecture that the probability that a particular vertex on the boundary of the diagonal half-plane percolation model is connected to no other vertices on the boundary is exactly 3/8. Is this amenable to Monte Carlo study? Not if the clusters are too big! Hence my question. | |
| May 4, 2010 at 1:26 | comment | added | James Propp | Thanks to Gil, Tom, and Leandro for their comments. I don't understand Tom's remark that $p_{\rm crit}$ will no longer be 1/2 for the diagonal half-plane model; can Tom (or someone else) explain this? In any case, when I wrote "critical edge-percolation" in my original posting, I meant that each edge is included with probability 1/2. The Nienhuis and Cardy conjectures, taken together, seem to suggest that the answer to Gil's question ("Are there reasons to believe that the behavior for the half plane is different than the behavior for the full plane?") is an emphatic "Yes". | |
| May 2, 2010 at 23:00 | answer | added | Tom LaGatta | timeline score: 4 | |
| May 2, 2010 at 21:43 | comment | added | Leandro | James, in the paper, Mean-field critical behaviour for percolation in high dimensions, math.bme.hu/~balint/oktatas/perkolacio/percolation_papers/… Hara and Slade mention on the page 337 that $$\mathbb P_{p_c}(\{|C(0)|=n\})\sim n^{-1-1/\delta}.$$ I did not remember if this asymptotic behavior it is expected only for high dimensions and if the invariance properties of $p_{uv}$ are required, anyway there is some references about this problem in the same page, maybe you can find something relevant for your problem there. | |
| May 2, 2010 at 20:42 | comment | added | Tom LaGatta | James, I agree with Gil in that it should probably be the same qualitative phenomenon as the full lattice. Of course, the critical percolation probability will no longer be 1/2 as on the full square lattice. What's your motivation for studying this question? | |
| May 2, 2010 at 5:15 | comment | added | James Propp | I have no reason to think that the behavior for the half plane is different than the behavior for the full plane, so I guess I should ask, by way of background, what the answer is for percolation in the full plane. | |
| May 2, 2010 at 4:54 | comment | added | Gil Kalai | Very nice question. Are there reasons to believe that the behavior for the half plane is different than the behavior for the full plane? | |
| May 2, 2010 at 4:45 | history | asked | James Propp | CC BY-SA 2.5 |