  The maximal (or Hilbertian) correlation coefficient between two random
variables X and Y, denoted by \{X:Y\}, is the supremum of the |Corr(f(X),g(Y))|
for real measurable functions f, g, where "Corr" denotes Pearson's correlation
coefficient. It is a classical result that for independent pairs of variables
(X_i,Y_i)_{i\in I}, \{\vec{X}_I:\vec{Y}_I\} is the supremum of the \{X_i:Y_i\}.
The main goal of this monograph is to prove similar tensorization results when
one only has partial independence between the (X_i,Y_i); more generally, for
random variables (X_i)_{i\in I}, (Y_j)_{j\in J}, we will look for bounds on
\{\vec{X}_I:\vec{Y}_J\} from bounds on the \{X_i:Y_j\}, i\in I, j\in J.
  My tensorization theorems will imply new decorrelation results for models of
statistical physics exhibiting asymptotic independence, like the subcritical
Ising model. I shall prove that for such models, two distant bunches of spins
are decorrelated (in the Hilbertian sense) uniformly in their sizes and shapes:
if I and J are two sets of spins such that dist(i,j)\geq d for all i\in I, j\in
J, then one gets a nontrivial bound for \{\vec{X}_I:\vec{Y}_J\} only depending
on d.
  Still for models like the subcritical Ising one, I shall also prove how
Hilbertian decorrelations may be used to get the spatial CLT or the (strict)
positiveness of the spectral gap for the Glauber dynamics, via tensorization
techniques again.
  Besides all that, I shall finally prove a new criterion to bound the maximal
correlation \{F:G\} between two \sigma-algebras F and G form a uniform bound on
the |P[A\cap B]-P[A]P[B]|/\sqrt{P[A]P[B]} for all A\in F, B\in G. Such criteria
were already known, but mine strictly improves those and can moreover be proved
to be optimal.
