  Let $X^N = (X_1^N,\dots, X^N_d)$ be a d-tuple of $N\times N$ independent GUE
random matrices and $Z^{NM}$ be any family of deterministic matrices in
$\mathbb{M}_N(\mathbb{C})\otimes \mathbb{M}_M(\mathbb{C})$. Let $P$ be a
self-adjoint non-commutative polynomial. A seminal work of Voiculescu shows
that the empirical measure of the eigenvalues of $P(X^N)$ converges towards a
deterministic measure defined thanks to free probability theory. Let now $f$ be
a smooth function, the main technical result of this paper is a precise bound
of the difference between the expectation of $$\frac{1}{MN}\text{Tr}\left(
f(P(X^N\otimes I_M,Z^{NM})) \right)$$ and its limit when $N$ goes to infinity.
If $f$ is six times differentiable, we show that it is bounded by
$M^2\left\Vert f\right\Vert_{\mathcal{C}^6}N^{-2}$. As a corollary we obtain a
new proof of a result of Haagerup and Thorbj\o rnsen, later developed by Male,
which gives sufficient conditions for the operator norm of a polynomial
evaluated in $(X^N,Z^{NM},{Z^{NM}}^*)$ to converge almost surely towards its
free limit. Restricting ourselves to polynomials in independent GUE matrices,
we give concentration estimates on the largest eingenvalue of these polynomials
around their free limit. A direct consequence of these inequalities is that
there exists some $\beta>0$ such that for any $\varepsilon_1<3+\beta)^{-1}$ and
$\varepsilon_2<1/4$, almost surely for $N$ large enough,
$$-\frac{1}{N^{\varepsilon_1}}\ \leq \| P(X^N)\| - \left\Vert P(x)\right\Vert
\leq\ \frac{1}{N^{\varepsilon_2}}.$$ Finally if $X^N$ and $Y^{M_N}$ are
independent and $M_N = o(N^{1/3})$, then almost surely, the norm of any
polynomial in $(X^N\otimes I_{M_N},I_N\otimes Y^{M_N})$ converges almost surely
towards its free limit. This result is an improvement of a Theorem of Pisier,
who was himself using estimates from Haagerup and Thorbj\o rnsen, where $M_N$
had size $o(N^{1/4})$.

Probability;Operator Algebras