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let us look at the function $N^d e^{-N}$, for each $d\in \mathbb{N}$. The graphs of the function for various values of $d$ show a striking phenomenon: the graph look parallel, and with a near-constant distance between each consecutive graphs.

enter image description here

Put in other words, Let $p<1$, and for each $d\in \mathbb{N}$ look at the $N$ that satisfies $N^de^{-N}=p$. Then it seems that the resulting function $N(d)$ is $o(n^2)$ or even $\Theta(n)$, and furthermore that for all $d$, $N(d+1)-N(d)$ is approximately uniform in $p$.

This phenomenon is observed here without explaining the exact argument. This also gives a good statistical motivation for asking the question.

What will be a rigorous statement of this phenomenon, and why is it true?

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    $\begingroup$ Note that the vertical axis has a log scale. He is effectively plotting $\ln(x^ne^{-x})$ for multiple values of $n$. Once $x$ is significantly bigger than $n$, the exponential decay dominates over the power term and since $\ln(e^{-x})$ is just $-x$ we get an apparently straight line. $\endgroup$ – almagest Apr 23 '16 at 20:01
  • $\begingroup$ @almagest Thanks for the nice observation. Does the supposed-linearity of $N(d)$ follow from it? (It is unaffected by whether we use a log-scale or not.) $\endgroup$ – Emolga Apr 24 '16 at 11:43
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For all $d \geq e$ and $0 < p < 1$ the equation

$$ N^d = pe^N $$

has two positive solutions $N$, call them $N_1$ and $N_2$. Since the maximum of $x \mapsto x^d e^{-x}$ occurs at $x=d$ and $x^d e^{-x} \to 0$ as $x \to \infty$ we know that $N_1 < d < N_2$.

We'll first investigate the behavior of $N_1$, then $N_2$ afterwards.

Because the maximum of $x \mapsto x^d e^{-x}$ is located at $x=d$, if $1 + \epsilon < x \leq d$ for some $\epsilon > 0$ then

$$ x^d e^{-x} > (1+\epsilon)^d e^{-1-\epsilon} \to \infty $$

as $d \to \infty$. In this case the equation $x^d e^{-x} = p$ will not hold for $d$ large enough. Consequently we may conclude that

$$ \limsup_{d \to \infty} N_1 \leq 1. $$

This is clearly not the solution we're looking for, so let's move on to $N_2$.

We know that $N_2 > d$. Taking logarithms of the original equation we have

$$ d \log N_2 = N_2 + \log p, \tag{$*$} $$

and so

$$ N_2 > d \log d - \log p. $$

In fact this captures the asymptotic behavior of $N_2$. Indeed, as $d \to \infty$ we know that $N_2 \to \infty$, so

$$ N_2 + \log p \sim N_2 $$

as $d \to \infty$, and thus equation $(*)$ implies

$$ d\log N_2 \sim N_2 $$

as $d \to \infty$. Taking logarithms again we see that

$$ \log d = \log N_2 - \log\log N_2 + o(1) $$

as $d \to \infty$. Dividing both sides by $\log N_2$ yields

$$ \frac{\log d}{\log N_2} = 1 - \frac{\log\log N_2}{\log N_2} + o(1) = 1 + o(1) $$

and thus $\log N_2 \sim \log d$. Plugging this back into $(*)$ allows us to conclude that

$$ N_2 \sim d\log d. $$

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