The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants

In the setting of The Wick-Square Problem on the Torus: Standing Notation, let $c_{k}$ be the free-field variances. Recall that $n\le3$ (The Wick-Square Problem on the Torus: Standing Notation §dimension). Then clauses 1 and 2 hold for every mode $k\in\mathbb{Z}^{n}$, and in clause 3 the bounds hold for every mode and the family is cube-summable.

1. (Root)¶ There is exactly one positive real number $q_{k}$ with

$$2q_{k}^{2}+(\gamma+2\mu_{k})\,q_{k}=\beta .$$

2. (Gaussian profile)¶

$$0\le\frac{\beta}{2\mu_{k}}-q_{k}\le\frac{\beta\gamma+\beta^{2}}{4\mu_{k}^{2}} .$$

3. (Renormalised constants)¶ The real number $a_{k}=\dfrac{\nu q_{k}-\beta c_{k}}{\gamma}$ satisfies

$$-\frac{\nu(\beta\gamma+\beta^{2})}{4\gamma\,\mu_{k}^{2}}\le a_{k}\le0,$$

and the family $k\mapsto a_{k}$ is cube-summable.