The Wall Energy of a Given Radius and the Wall Force of a Noncommutative Law

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $R>0$ be real. For $j\in[d]$ and $n\in\mathbb{N}$, $x_{j}^{n}$ is the product of $n$ factors $x_{j}$, that is, the product along the word of length $n$ all of whose letters are $1$, for the $1$-tuple $(x_{j})$; natural numbers appearing as coefficients below are read as real numbers, $R^{n}$ is the natural power and $R^{-n}$ its multiplicative inverse. For $n\in\mathbb{N}$ and $\lambda\in\Sigma_{d}$ let

$$E^{R}_{n}(\lambda)=\sum_{j=1}^{d}\sum_{k=1}^{n}R^{-2k}\,\lambda(x_{j}^{2k}),$$

and for $n\in\mathbb{N}$ and $j\in[d]$ let

$$F^{R}_{n,j}=\sum_{k=1}^{n}2k\,R^{-2k}\,x_{j}^{2k-1}\in\mathcal{P}_{d}.$$

Each $\lambda(x_{j}^{2k})$ is real and nonnegative by condition (b) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state, since $x_{j}^{2k}=x_{j}^{k}x_{j}^{k}$ by Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §concatenation and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, and $(x_{j}^{k})^{*}=x_{j}^{k}$ by Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §reversal (a word all of whose letters are $1$ is its own reversal), Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint (which also gives $x_{j}^{*}=x_{j}$), Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §adjoint.

1. (Domain)¶ $\mathcal{D}_{R}$ is the set of $\lambda\in\Sigma_{d}$ for which the real sequence $(E^{R}_{n}(\lambda))_{n\in\mathbb{N}}$ is bounded above.

2. (Wall energy)¶ For $\lambda\in\mathcal{D}_{R}$ the sequence $(E^{R}_{n}(\lambda))_{n\in\mathbb{N}}$ is nondecreasing, as $E^{R}_{n+1}(\lambda)-E^{R}_{n}(\lambda)=\sum_{j=1}^{d}R^{-2(n+1)}\lambda(x_{j}^{2(n+1)})\ge0$, and bounded above, so it converges by A Bounded Monotone Sequence of Real Numbers Converges §nondecreasing. Its limit $\mathcal{W}_{R}(\lambda)$ is the wall energy of radius $R$ of $\lambda$.

3. (Wall force)¶ $\lambda\in\Sigma_{d}$ has a square-integrable wall force of radius $R$ if for every $j\in[d]$ the sequence $(\widehat{F^{R}_{n,j}})_{n\in\mathbb{N}}$ of classes in the complex GNS space $\mathcal{H}_{\lambda}$ converges. Its limit is written $F^{R}_{j}(\lambda)$, and $F^{R}(\lambda)=(F^{R}_{1}(\lambda),\dots,F^{R}_{d}(\lambda))\in\mathcal{H}_{\lambda}^{d}$ is the wall force of $\lambda$.