Daido order parameters

PDF p. 11: Section 2.1, equation (2.7)

Let $V$ be a nonempty finite vertex set, put $n=\mathrm{\mid }V\mathrm{\mid }$, and let $\theta \in T_V={\mathbb{R}}^V\mathrm{/}(2\pi \mathbb{Z}{)}^V$ be a phase configuration. Thus each phase ${\theta }_x$ is a real angle modulo $2\pi$; a lift is a real representative of that phase. For each integer $k\ge 1$, the $k$th Daido order parameter is the complex number

${\rho }_k(\theta ):=\frac{1}{\mathrm{\mid }V\mathrm{\mid }}{\sum }_{x\in V}{e}^{ik{\theta }_x},$

where the exponential is evaluated using any real lift of each phase. Here $e^{iu}=\cos u+i\sin u$ for real $u$, with $i^2=-1$.

Given: A nonempty finite set $V$, a phase configuration $\theta \in T_V$, and an integer $k\ge 1$.

Goal: Show that the displayed formula determines a unique complex number independently of the lifts.

1.1 If a lift of ${\theta }_x$ is replaced by another, [F1] makes the change $2\pi m_x$ for an integer $m_x$. The argument $k{\theta }_x$ then changes by $2\pi (k{m}_x)$, so [F2] leaves the corresponding exponential unchanged. [F1, F2, given, algebra]

2.1 By step 1.1 every summand is independent of its lift. The sum has finitely many complex summands and $\mathrm{\mid }V\mathrm{\mid }=n$ is positive, so its average is a uniquely determined complex number. This establishes the definition. [step 1.1, given, algebra] ∎

The name and formula are attributed to Expander graphs are globally synchronizing, PDF page 11, Section 2.1, equation (2.7). The justification above uses the stated mathematical suppliers.