Description
Bécs-Budapest Valószínűségszámítási Szeminárium
Abstract: We consider long range percolation on Z^d in a finite box of volume n in dimension d. In this model edges are independently present between any tow vertices x,y with probability constant times 1/|x-y|^(alpha d). When the multiplying constant is large enough, the model is supercritical, and there is an infinite component of density theta = Prob( the origin is part of the infinite component); which then intuitively suggests the existence of a giant component in a finite box; having density roughly theta. In this talk we study large deviation for this giant component; and show that it undergoes a dichotomy: the lower tail Prob( largest component < n (theta-epsilon)) has polynomial speed in the volume while the upper tail Prob( largest component > n (theta+epsilon) ) has linear speed. If one modifies the model to allow for inhomogeneous vertex marks, (e.g. as in geometric inhomogeneous random graphs or the Poisson Boolean model with random radii); then the lower tail remains to have polynomial speed, but the upper tail becomes much heavier. The speed is `log n'; and we can express the exact rate function using the (inverse of the) generating function of the cluster of the origin in the infinite model. Joint work with Joost Jorritsma and Dieter Mitsche.
A részletes program:
https://sites.google.com/view/budapest-vienna-proba-semi/home