We first recall some basics about Lawlor necks [51][45], which are non-compact embedded exact special Lagrangians $L_{\phi,A}$ inside the standard Euclidean $\mathbb{C}^{n}$, asymptotic at infinity to the union of two planes

$$\Pi_{0}=\mathbb{R}^{n},\quad\Pi_{\phi}=(e^{i\phi_{1}},\ldots e^{i\phi_{n}})\mathbb{R}^{n},\quad 0<\phi_{i}<\pi,\quad\sum\phi_{i}=\pi.$$

Symplectic topologically, they can be viewed as a realisation of the Lagrangian handle $S^{n-1}\times\mathbb{R}$ that appears in the Lagrangian connected sum construction. This motivates the ansatz

$$L_{\phi,A}=\{(z_{1}(y)x_{1},...,z_{n}(y)x_{n}):y\in\mathbb{R},x_{k}\in\mathbb{R},x^{2}_{1}+\ldots+x_{n}^{2}=1\}.$$

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The special Lagrangian condition translates into an ODE system on the functions $z_{1}(y),\ldots z_{n}(y)$, which can be solved exactly as follows.

Let $n>2$ and $a_{1},...,a_{n}>0$, and define polynomials $p,P$ by

$$p(x)=(1+a_{1}x^{2})\ldots(1+a_{n}x^{2})-1,\quad P(x)=\frac{p(x)}{x^{2}}.$$

Define real numbers $\phi_{1},...,\phi_{n}$ and $A$ by

$$\phi_{k}=a_{k}\int_{-\infty}^{\infty}\frac{dx}{(1+a_{k}x^{2})\sqrt{P(x)}},\quad A=\int_{-\infty}^{\infty}\frac{dx}{2\sqrt{P(x)}}$$

Clearly $\phi_{k},A>0$, and elementary integration shows $\sum\phi_{i}=\pi$. This yields a 1-1 correspondence between $n$-tuples $(a_{1},\ldots,a_{n})$ with $a_{k}>0$, and $(n+1)$-tuples $(\phi_{1},\ldots,\phi_{n},A)$ with $\phi_{k}\in(0,\pi)$, $\sum\phi_{k}=\pi$ and $A>0$. Setting

$$z_{k}(y)=e^{i\psi_{k}(y)}\sqrt{a_{k}^{-1}+y^{2}},\quad\text{where }\psi_{k}(y)=a_{k}\int_{-\infty}^{y}\frac{dx}{(1+a_{k}x^{2})\sqrt{P(x)}},$$

yields the solution $(z_{1}(y),\ldots,z_{n}(y))$, hence the Lawlor necks $L_{\phi,A}$.

For fixed asymptotic planes $\Pi_{0},\Pi_{\phi}$, the Lawlor necks arise in a 1-parameter family, related to each other by the rescaling $\vec{z}\to\lambda\vec{z}$ in $\mathbb{C}^{n}$, and $A$ behaves like 2-dimensional area $A\to\lambda^{2}A$ under this scaling. One also observes that asymptotically near infinity, the Lawlor necks are graphs over $\Pi_{0}$ (resp. $\Pi_{\phi}$) of the differential $df$, where

$$|f|=O(|\vec{x}|^{2-n}),\quad|\partial^{k}f|=O(|\vec{x}|^{2-n-k}).$$

We say the Lawlor neck has asymptotic decay rate $2-n$. The upshot is that it approaches $\Pi_{0}\cup\Pi_{\phi}$ sufficiently fast.