User yogesh - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T13:52:59Z http://mathoverflow.net/feeds/user/11409 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81840/negatively-associated-point-processes Negatively associated point processes yogesh 2011-11-24T22:59:12Z 2011-11-25T16:50:56Z <p>A point process $\Phi$ is said to be negatively associated if for any finitely many bounded Borel subsets $B_1,B_2,...,B_n,$ we have that </p> <p>$$Cov( f(\Phi(B_1),\ldots,\Phi(B_l))g(\Phi(B_{l+1}),\ldots,\Phi(B_{n}))) \leq 0$$ </p> <p>whenever $f,g$ are non-decreasing, non-negative functions and $(B_1 \cup ... B_l) \cap (B_{l+1} \cup ... B_n) = \emptyset.$ Here $\Phi(B_i)$ denotes the random number of points within the set $B_i$.</p> <p>This is an extension of the notion of <a href="http://www.springerlink.com/content/j35n111725n73h44/" rel="nofollow">positive association for random measures</a>. There are <a href="http://projecteuclid.org/euclid.aop/1176992810" rel="nofollow">many examples of positively associated point processes</a>. A <a href="http://arxiv.org/abs/math/0404095" rel="nofollow">detailed study of Negative association</a> by R. Pemantle in 2000 led to further interest in the concept of negative dependence. There are many examples of such measures in the discrete setting but i am not aware of any such example of spatial point processes i.e, point processes in $R^d$. </p> <p>In a <a href="http://arxiv.org/abs/math/0204325" rel="nofollow">paper of R. Lyons</a>, it is proved that discrete determinantal probability measures are negatively associated. Though i expect it to work for spatial determinantal processes as well, currently it stands unproved. </p> <p>Is someone aware of any negatively associated point process ? Or do you have suggestion for any other point process that you suspect will be negatively associated ? </p> http://mathoverflow.net/questions/79321/computational-topology-paper/81838#81838 Answer by yogesh for Computational Topology Paper yogesh 2011-11-24T22:10:43Z 2011-11-24T22:10:43Z <p>The interesting books i know of are Edelsbrunner/Harer and Zomordian's thesis. On similar topics, the <a href="http://comptop.stanford.edu/" rel="nofollow">Comtop group</a> at Stanford has very detailed information. </p> http://mathoverflow.net/questions/40679/is-there-a-percolation-threshold-in-the-hard-discs-model/52132#52132 Answer by yogesh for Is there a percolation threshold in the hard discs model? yogesh 2011-01-15T00:05:51Z 2011-01-15T00:05:51Z <p>Perhaps you are already aware of asymptotics for component sizes in the continuum percolation model. It is there in the book of M. Penrose.</p> <p>Since you asked about percolation on repulsive point processes, here is a <a href="http://arxiv.org/abs/1009.5696" rel="nofollow">compressed version</a> of our results of non-trivial phase transition on sub-Poisson point processes , point processes that are less clustered than a Poisson in a certain sense. Our main example is a perturbed lattice. We also show that stationary determinantal point processes have a non-zero critical intensity as regards percolation. I think these point processes can be considered as repulsive point processes.</p> http://mathoverflow.net/questions/39491/a-point-process-is-characterized-by-its-void-probabilities/48653#48653 Answer by yogesh for a point process is characterized by its void probabilities yogesh 2010-12-08T16:16:23Z 2010-12-08T16:16:23Z <p>To add to Omer's concise explanation, the general result is known as Choquet's capacity theorem. It says that the void probabilities characterise any random closed set. Simple point processes are an example of random closed sets. </p> http://mathoverflow.net/questions/81840/negatively-associated-point-processes Comment by yogesh yogesh 2011-11-25T16:59:19Z 2011-11-25T16:59:19Z Thanks Ori. I am puzzled as to why i missed this earlier. In fact, one might be able to show that superposition of independent NA point processes will lead to a NA point process. So, a point process with $n$ uniformly distributed points in each cube of the $Z^d$ lattice will be negatively associated point process on the entire plane.