Consider a model gas of electrically charged particles in D a bounded subset of Rd. The interaction
given by ijV (jxi ? xj j) where i; j 2 f?1; 1g stands for the particle charge and xi; xj 2 D for the
particle position and V represents the eletrical potential (which is a function of the distance). Given
and two positive parameters, our model is described by the following partition function which induces
a probability distribution on the set of charged particles
ZV
; (O) =
1X
n=0
n
n!
X
()ni
=12f?1;1gn
Z
Dn
exp
?
X
1i<jn
iiV (jzi ? zj j)
Yn
i=1
dzi
We are interested speci cally in the case when V diverges logarithmically in zero V (r) = ?log r + O(1).
In that case, we observe that the above partition function is nite if and only if d. The aim of this
talk is to introduce renormalization techniques using connection with Gaussian elds which allows for a
de nition of the Log-gas when 2 [d; 2d). (joint work with V.Vargas and R.Rhodes).
