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Oct
6
comment Onsager's Regression Hypothesis, Explained and Demonstrated
Dear John, Many thanks!!
Oct
5
comment Onsager's Regression Hypothesis, Explained and Demonstrated
Many thanks, John. (For me (given my own lack of background) an even more elementary/sef-contained/mathematical answer could be perhaps even more accesible/helpful.)
Oct
4
comment Sampling typical clusters between distant points in subcritical percolation
Actually, maybe we can use the above idea iteratively. We can sample fairly efficiently condition on the case that there is left to right crossing for the T by 2M rectangles. Given this we can try being more ambitious and try to sample from the results the cases where there is crossing in 2T by 2M rectangles, and then (say) by 4T by M rectangles. This may lead to a not too ineficient method. (We can tune it further by letting M depend on the X coordinate, and perhaps also by splitting the rectangles horizontally as well.
Oct
4
comment Sampling typical clusters between distant points in subcritical percolation
Dear Yvan, Indeed, maybe I was overly skeptical. Suppose you want to sample a random such cluster assuming there is a path between (0,0) to (N,0) with y-coordinate in -M,M. Then you can condition on having a left-to-right crossings in T by 2M rectangles of the form [rT,(r+1)T] X [-M,M]. These events are independent so you can sample conditioned to all of them satisfied. Having a path from (0,0) to (N,0) will still be rare conditioned on the existence of these crossings but not as rare. Thanks for the reference to your work.
Oct
3
comment Sampling typical clusters between distant points in subcritical percolation
"As an example, he's sampling 18 disjoint crossing clusters for critical percolation on $Z^3$ in a box of size 128×128×2000 , an event of probability of order $10^{āˆ’300}$". Yvan, I see very little hope that this sampling is related to the distribution claimed to be sampled. (This reminds me the following joke: A person says he what to sell his dog, "for how much?" his friends ask, "for $10^{300}$ dollars" he says. The next day they ask him if he was successful. "Yes!, I sold it for 10 kittens worth $10^{299}$ dollars each!")
Oct
3
comment Sampling typical clusters between distant points in subcritical percolation
"one has a lot of information about the geometry of the corresponding cluster (e.g., the corresponding cluster converges to a brownian bridge as Nā†’āˆž under diffusive scaling, it has a maximal "width" of order logN , etc.)." Perhaps the best bet is to understand where this information comes from and do the proofs suggest how a random cluster of the kind you want looks like. Also, you can use the known information to check suggestions for such samplings like the one in the editted part of the question.
Oct
2
comment Sampling typical clusters between distant points in subcritical percolation
Dear Yvan, I DONT refer here to planar duality. In fact duality between the behavior of events in the subcritical and supercritical probabilities exists for various models e.g. the Erdos Renyi model of random graph. For example, we may wonder if the distribution of clusters containing (0,0) and (N,0) in subcritical percolation with edge probability p is related to the distribution of the cluster containing (0,0) and (N,0) conditioned on the existence of such cluster and it not being infinite. (Such a relation is a bit wild speculation but I see no other hope for an efficient algorithm.)
Oct
2
answered Sampling typical clusters between distant points in subcritical percolation
Sep
26
asked Onsager's Regression Hypothesis, Explained and Demonstrated
Sep
24
comment The Role of Rigor
Moshe, in any case I will be very interested to see specific answers (not discussions) on what notions/results from physics still lack mathematical rigorous description; on cases where rigorous mathematical framework came from advances in physics and where rigorous mathematical framework proposed in mathematics was useful in physics.
Sep
24
comment The Role of Rigor
Dear Moshe, By no mean did I mean to belittle the quality of discussions on blogs and popular books on this matter. Some such discussions are of good quality! Part 5 of my question was just asking for studies of this issue on professonal academic journals. Parts 1-4 are questions (hopefully answerable) aimed at mathematical physicists and theoretical physcisits. Also, I completely agree with not being sure if the question is within the purview of the forum.Dear Michael, Thanks! I had also another MO question in mind mathoverflow.net/questions/37610/…