 
Summary: Submitted to the Annals of Probability
HIGH LEVEL EXCURSION SET GEOMETRY FOR
NONGAUSSIAN INFINITELY DIVISIBLE RANDOM
FIELDS
By Robert J. Adler Gennady Samorodnitsky,
and Jonathan E. Taylor,
Technion, Cornell and Stanford.
We consider smooth, infinitely divisible random fields X(t), t
M , M Rd
, with regularly varying L´evy measure, and are inter
ested in the geometric characteristics of the excursion sets
Au = t M : X(t) > u
over high levels u.
For a large class of such random fields we compute the u
asymptotic joint distribution of the numbers of critical points, of vari
ous types, of X in Au, conditional on Au being nonempty. This allows
us, for example, to obtain the asymptotic conditional distribution of
the Euler characteristic of the excursion set.
In a significant departure from the Gaussian situation, the high
level excursion sets for these random fields can have quite a compli
