4 The master inequality: for polynomials

In this chapter, we prove the master inequality I, as done in [ . This will require some careful analysis on polynomials. Throughout the rest of this preprint, the set of real polynomials will be denoted as \(\mathcal{P}\), and the set of real polynomials of degree at most \(q\) by \(\mathcal{P}_q\).

4.1 Markov inequalities

Proposition 4.1.1

[todo]

4.2 Non-commutative polynomials with matrix coefficients

Definition 4.2.1

Let \(D \geq 1\) and \(r \geq 1\). A non-commutative polynomial with matrix coefficients \(P \in \mathcal{M}_D(\mathbb {C}) \otimes \mathbb {C}\langle x_1, \ldots , x_{r} \rangle \) is an expression of the form \(\sum _{w \in \mathbf{W}_r} A_w \otimes w(x_1, \ldots , x_{r})\).

Definition 4.2.2

Let \(P = \sum _{w \in \mathbf{W}_r} A_w \otimes w(x_1, \ldots , x_{r})\in \mathcal{M}_D(\mathbb {C}) \otimes \mathbb {C}\langle x_1, \ldots , x_{r} \rangle \). For \((U_1, \ldots , U_r)\) be a \(r\)-tuple of unitary matrices of shared dimension, we define \(P(U_1, \ldots , U_r) = \sum _{w \in \mathbf{W}_r} A_w \otimes w(U_1, \ldots , U_r)\) using the word map defined in Definition 3.3.3.

Definition 4.2.3

[todo]

Definition 4.2.4

We define a norm on \(\mathcal{M}_D(\mathbb {C}) \otimes \mathbb {C}\langle x_1, \ldots , x_{r} \rangle \) by taking for a non-commutative polynomial with matrix coefficients \(P \in \mathcal{M}_D(\mathbb {C}) \otimes \mathbb {C}\langle x_1, \ldots , x_{r} \rangle \),

\begin{equation*} \| P\| _{D,r} := \sup \{ \| P(U_1, \ldots , U_r)\| , (U_i)_{1 \leq i \leq r} \text{ unitary matrices}\} . \end{equation*}

4.3 The Master inequality for polynomials

We are now ready to prove the following statement.

Theorem 4.3.1

Let \(r \geq 2\) and \(D \geq 1\). Fix a self-adjoint non-commutative polynomial \(P \in \mathcal{M}_D(\mathbb {C}) \otimes \mathbb {C}\langle x_1, \ldots , x_{r} \rangle \) of degree \(q_0\), and let \(K = \| P\| _{D,r}\). There exists a family of linear functionals \(\nu _i : \mathcal{P} \rightarrow \mathbb {C}\) for \(i \geq 0\) such that for every \(N,m,q \geq 1\), and any \(h \in \mathcal{P}_q\),

\begin{equation*} \left|\mathbb {E}_{N,r}\left[\mathrm{tr}_{ND} h(P(S_1, \ldots , S_r))\right] - \sum _{i=0}^{m-1} \frac{\nu _i(h)}{N^i}\right| \leq \frac{(4 q q_0(1+\log r))^{4m}}{N^m} \| h\| _{C^0([-K,K])}. \end{equation*}

This will be done in several steps.

4.3.1 Proof for large values

We first prove the result for large \(N\), which is the main challenge.

Lemma 4.3.2

[todo]

4.3.2 Bound on the coefficients

We now prove a bound on the coefficients of the expansion.

Lemma 4.3.3

With the notations of Theorem 4.3.1, for every \(m, q \geq 1\) and \(h \in \mathcal{P}_q\),

\begin{equation*} |\nu _m(h)| \leq (4 q q_0(1+\log r))^{4m} \| h\| _{C^0([-K,K])}. \end{equation*}

This relies on the following simple lemma on asymptotic expansions.

Lemma 4.3.4

Let \((F_N)_{N \geq 0}\) be a sequence. We assume there exists coefficients \((\nu _i)_{i \geq 0}\), constants \((C_m)_{m \geq 1}\) and \((M_m)_{m \geq 1}\) such that, for all \(m \geq 1\), all \(N \geq M_m\),

\begin{equation*} \left| F_N - \sum _{i=0}^{m-1} \frac{\nu _i}{N^i} \right| \leq \frac{C_m}{N^m}. \end{equation*}

Then, for all \(m \geq 1\), \(|\nu _m| \leq C_m\).

Proof ▶

Let \(m \geq 1\) and \(N \geq \max (M_m, M_{m+1})\). We have

\begin{equation*} \nu _m = N^m \left(F_N - \sum _{i=0}^{m-1} \frac{\nu _i}{N^i} \right) - N^m \left(F_N - \sum _{i=0}^{m} \frac{\nu _i}{N^i}\right) \end{equation*}

so by the triangle inequality,

\begin{equation*} |\nu _m| \leq N^m \left|F_N - \sum _{i=0}^{m-1} \frac{\nu _i}{N^i} \right| + N^m \left|F_N - \sum _{i=0}^{m} \frac{\nu _i}{N^i}\right|. \end{equation*}

We apply the hypothesis to \(m\) and \(m+1\) (which we can as \(N \geq M_m\) and \(N \geq M_{m+1}\)), which yields

\begin{equation*} |\nu _m| \leq N^m \frac{C_m}{N^m} + N^m \frac{C_{m+1}}{N^{m+1}} = C_{m} + \frac{C_{m+1}}{N}. \end{equation*}

This is true for all \(N \geq \max (M_m,M_{m+1})\), and hence we can take the limit \(N \rightarrow + \infty \), and obtain \(|\nu _m| \leq C_m\).

We will also need to bound the \(0\)-th coefficient.

Lemma 4.3.5

For any \(h \in \mathcal{P}\),

\begin{equation*} \nu _0(h) = \lim _{N \rightarrow +\infty } \mathbb {E}_{N,r}[\mathrm{tr}_{ND} h(P(S_1, \ldots , S_r))]. \end{equation*}

Proof ▶

Let \(h \in \mathcal{P}\) and take \(q \geq 1\) such that \(h \in \mathcal{P}_q\). We apply Lemma 4.3.2 for \(m=1\). Hence, for any \(N \geq M\),

\begin{equation*} \left| \mathbb {E}_{N,r}[\mathrm{tr}_{ND} h(P(S_1, \ldots , S_r))] - \nu _0(h) \right| \leq \frac{(4 q q_0(1+\log r))^4}{N} \end{equation*}

The right-hand side goes to \(0\) as \(N \rightarrow \infty \), which leads to the claim.

Lemma 4.3.6

For any \(h \in \mathcal{P}\), any \(N \geq 1\),

\begin{equation*} |\mathbb {E}_{N,r}[\mathrm{tr}_{ND} h(P(S_1, \ldots , S_r))]| \leq \| h\| _{C^0([-K,K])}. \end{equation*}

todo ▶

We can finally prove Lemma 4.3.3.

Proof ▶

Let \(m \geq 0\), \(q \geq 1\) and \(h \in \mathcal{P}_q\).

If \(m=0\), by Lemma 4.3.5,

\begin{equation*} \nu _0(h) = \lim _{N \rightarrow +\infty } \mathbb {E}_{N,r}[\mathrm{tr}_{ND} h(P(S_1, \ldots , S_r))]. \end{equation*}

By Lemma 4.3.6, this sequence is bounded by \(\| h\| _{C^0([-K,K])}\), and in particular we get that at the limit \(|\nu _0(h)| \leq \| h\| _{C^0([-K,K])}\). This is the claim for \(m=0\).

The result for \(m \geq 1\) is a direct combination of Lemma 4.3.2 and Lemma 4.3.4.

4.3.3 Proof for small values

Let us now conclude the proof of Theorem 4.3.1.

Proof ▶

[todo]

4.4 Special cases

Lemma 4.4.1

With the notations of Theorem 4.3.1, if \(h\) is a constant polynomial, then

\(\nu _0(h)=h(0)\);
for \(i {\gt}0\), \(\nu _i(h)=0\).

Proof ▶

If \(h\) is constant, then for any \(N \geq 1\), \(\mathbb {E}_{N,r}\left[\mathrm{tr}_{ND} h(P(S_1, \ldots , S_r))\right] = h(0)\) and the sequence from the claim satisfies the inequality from Theorem 4.3.1 and is hence the value of \((\nu _i(h))_{i \geq 0}\) by uniqueness [todo].