  Let G be a group and V a finite dimensional representation of G over an
algebraically closed field k of characteristic p>0. Let $d_n(V)$ be the number
of indecomposable summands of $V^{\otimes n}$ of nonzero dimension mod p. It is
easy to see that there exists a limit $\delta(V):=\lim_{n\to
\infty}d_n(V)^{1/n}$, which is positive (and $\ge 1$) iff V has an
indecomposable summand of nonzero dimension mod p. We show that in this case
the number $$ c(V):=\liminf_{n\to \infty} \frac{d_n(V)}{\delta(V)^n}\in [0,1]
$$ is strictly positive and $$ \log (c(V)^{-1})=O(\delta(V)^2), $$ and moreover
this holds for any symmetric tensor category over k of moderate growth.
Furthermore, we conjecture that in fact $$ \log(c(V)^{-1})=O(\delta(V)) $$
(which would be sharp), and prove this for p=2,3; in particular, for p=2 we
show that $c(V)\ge 3^{-\frac{4}{3}\delta(V)+1}$. The proofs are based on the
characteristic p version of Deligne's theorem for symmetric tensor categories
obtained in earlier work of the authors. We also conjecture a classification of
semisimple symmetric tensor categories of moderate growth which is interesting
in its own right and implies the above conjecture for all $p$, and illustrate
this conjecture by describing the semisimplification of the modular
representation category of a cyclic p-group. Finally, we study the asymptotic
behavior of the decomposition of $V^{\otimes n}$ in characteristic zero using
Deligne's theorem and the Macdonald-Mehta-Opdam identity.
