Janson's height and width conjectures for simply generated trees

Let ${\mathrm{w}}=(w_k,k \ge 0)$ be a weight sequence with $w_0>0$ and with $w_k>0$ for some $k \ge 2$. For $n \ge 0$ with $Z_n({\mathrm{w}})>0$, let ${\mathcal{T}}_n$ be a simply generated tree of size $n$ with weight sequence ${\mathrm{w}}$. The parameters $\nu$ and $\sigma^2$ are defined from the generating function of the weight sequence as in the setup above. Janson's height and width conjectures. Along integers $n$ such that $Z_n({\mathrm{w}})>0$, as $n\to\infty$, the following hold: (1) if $\nu=1$ and $\sigma^2=\infty$, then

$$\frac{{\mathrm{ht}}({\mathcal{T}}_n)}{n^{1/2}}\to 0$$

in probability; (2) if $\nu=1$ and $\sigma^2=\infty$, then

$$\frac{{\mathrm{wid}}({\mathcal{T}}_n)}{n^{1/2}}\to\infty$$

in probability; (3) if $\nu<1$, then

$$\frac{{\mathrm{ht}}({\mathcal{T}}_n)}{n^{1/2}}\to 0$$

in probability; and (4) if $\nu<1$, then

$$\frac{{\mathrm{wid}}({\mathcal{T}}_n)}{n^{1/2}}\to\infty$$

in probability. These assertions concern simply generated trees in the infinite-variance and condensation regimes, predicting subdiffusive height and superdiffusive width. The paper addresses these conjectures, so their resolution should be checked against the results stated elsewhere in the source.