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Large deviations of the size of the giant component in boxes for long range percolation and other spatial random graphs

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 two 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). This suggests the existence of a giant component in a finite box of volume n; 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 in n. 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 becomes 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. The lower tail generally has a deep connection with the size of the second largest component in the box and also to the probability that the origin is in a finite component of size at least k in the infinite model. Joint work with  Dieter Mitsche.

Júlia Komjáthy

2024.06.24 15.00 GMT+2 (15.00 CEST)



Joost Jorritsma

2024.06.24 16.00 GMT+2 (15.00 CEST)

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