Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Percolation in a 2D Ising model

+ 5 like - 0 dislike
991 views

For a 2D Ising model with zero applied field, it seems logical to me that the phases above and below T_c will have different percolation behaviour. I would expect that percolation occurs (and hence there is a spanning cluster) below T_c and that it does not occur (no spanning clusters) above T_c. I'm looking for conformation that this is true, but haven't found anything so far. Is my conjecture correct?

This post imported from StackExchange Physics at 2014-06-06 02:40 (UCT), posted by SE-user Matthew Matic
asked May 4, 2012 in Theoretical Physics by Matthew Matic (25 points) [ no revision ]
Yes your conjecture is right. The Ising model is connected to the bond-correlated percolation model. See this paper.

This post imported from StackExchange Physics at 2014-06-06 02:40 (UCT), posted by SE-user Vijay Murthy
Thanks, though my suspicions were that it is related to site percolation. Specifically, I would expect that the spin configuration of the high temperature Ising model can be mapped to the configuration of sites for p \approx 1/2 bond percolation. I suppose that the bond-correlated percolation mapping also works at finite temperature.

This post imported from StackExchange Physics at 2014-06-06 02:40 (UCT), posted by SE-user Matthew Matic
@VijayMurthy it would be great if you post that as an answer (and just add a few words about the argument given in the paper).

This post imported from StackExchange Physics at 2014-06-06 02:40 (UCT), posted by SE-user David Z

1 Answer

+ 2 like - 0 dislike

You conjecture is correct. One can relate the 2d Ising model with the bond correlated percolation model. The details are in the paper Percolation, clusters, and phase transitions in spin models.

The basic idea is to consider interacting (nearest neighbor) spins as forming a bond with a certain probability. One can then show that the partition function of the Ising model is related to the generating function of the bond-correlated percolation model.

The above paper demonstrates that the bond-correlated percolation model has the same critical temperature and critical exponents as the 2d Ising model. However, the values of $T_c$ and the critical exponents seem to be dependent on exactly how one defines a bond. See section III.A.1 in Universality classes in nonequilibrium lattice systems (or arxiv version).

Nonetheless your intuitive picture that there would be spanning clusters below $T_c$ and no such clusters above $T_c$ remains valid.

EDIT 21 May 2012

I found a pedagogical paper that discusses this issue.

This post imported from StackExchange Physics at 2014-06-06 02:40 (UCT), posted by SE-user Vijay Murthy
answered May 5, 2012 by Vijay Murthy (90 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\varnothing$ysicsOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...