This post is about some aspects of the exponential series

$\mathrm{e}_N(z):=\sum_{\ell=0}^{N-1}\frac{z^\ell}{\ell!}.$

A bit of probabilistic intuition suggests that when ${z=r>0}$, the behavior of ${\mathrm{e}^{-rN}\mathrm{e}_N(Nr)}$ as ${N\rightarrow\infty}$ has a critical value ${r=1}$. Namely, if ${X_1,\ldots,X_N}$ are iid random variables following the Poisson distribution of mean ${r}$, then

$\mathrm{e}^{-rN}\mathrm{e}_N(Nr) =\mathbb{P}(X_1+\cdots+X_N<N) =\mathbb{P}\left(\frac{X_1+\cdots+X_N}{N}<1\right).$

Now as ${N\rightarrow\infty}$, the right hand side tends to ${0}$ if ${r>1}$ and to ${1}$ if ${r<1}$, since ${\lim_{N\rightarrow\infty}\frac{X_1+\cdots+X_N}{N}=r}$ almost surely by the law of large numbers. We have shown that

$\lim_{N\rightarrow\infty}\mathrm{e}^{-rN}\mathrm{e}_N(Nr)=\mathbf{1}_{r\leq 1}$

for any ${r\neq 1}$, but the convergence to this indicator does not hold for ${r=1}$ since by the central limit theorem, in this case,

$\mathbb{P}(X_1+\cdots+X_N<N) =\mathbb{P}\left(\frac{X_1+\cdots+X_N-N}{\sqrt{N}}<0\right) \rightarrow\frac{1}{2}.$

The same argument works with the Gamma distribution: namely, for any ${r>0}$,

$\frac{1}{(N-1)!}\int_r^\infty t^{N-1}\mathrm{e}^{-t}\,\mathrm{d}t =\mathrm{e}^{-r}\sum_{\ell=0}^{N-1}\frac{r^\ell}{\ell!},$

in other words, if ${Y_1,\ldots,Y_N}$ are iid random variables with law ${\mathrm{Gamma}(1,1)}$, then

$\mathbb{P}(Y_1+\cdots+Y_N>r)=\mathrm{e}^{-r}\mathrm{e}_N(r).$

Bayesian statisticians are quite familiar with these Gamma-Poisson games.

Error. For every ${N\geq1}$ and ${z\in\mathbb{C}}$,

$|\mathrm{e}_N(Nz)-\mathrm{e}^{Nz}\mathbf{1}_{|z|\leq1}|\leq r_N(z)$

where

$r_N(z):= \frac{\mathrm{e}^N}{\sqrt{2\pi N}}|z|^N\left(\frac{N+1}{N(1-|z|)+1} \mathbf{1}_{|z|\leq1} +\frac{N}{N(|z|-1)+1}\mathbf{1}_{|z|>1}\right).$

In particular, if we define

$\varphi^{1,N}(z):=\frac{\mathrm{e}^{-N|z|^2}}{\pi}\mathrm{e}_N(N|z|^2)$

then, for any compact subset ${K\subset\mathbb{C}\setminus\{z\in\mathbb{C}:|z|=1\}}$,

$\lim_{N\rightarrow\infty}\sup_{z\in K} \left|\varphi^{1,N}(z)-\frac{\mathbf{1}_{|z|\leq1}}{\pi}\right| = \pi^{-1}\lim_{N\rightarrow\infty}\sup_{z\in K} \left|\mathrm{e}^{-N|z|^2}\mathrm{e}_N(N|z|^2)-\mathbf{1}_{|z|\leq1}\right| =0.$

Proof. Following Mehta, for every ${N\geq1}$, ${z\in\mathbb{C}}$, if ${|z|\leq N}$ then

$\left|\mathrm{e}^{z}-\mathrm{e}_N(z)\right| =\left|\sum_{\ell=N}^\infty\frac{z^\ell}{\ell!}\right| \leq\frac{|z|^N}{N!}\sum_{\ell=0}^\infty\frac{|z|^\ell}{(N+1)^\ell} =\frac{|z|^N}{N!}\frac{N+1}{N+1-|z|},$

while if ${|z|>N}$ then

$|\mathrm{e}_N(z)| \leq \sum_{\ell=0}^{N-1}\frac{|z|^\ell}{\ell!} \leq \frac{|z|^{N-1}}{(N-1)!}\sum_{\ell=0}^{N-1}\frac{(N-1)^{\ell}}{|z|^\ell} \leq\frac{|z|^{N-1}}{(N-1)!}\frac{|z|}{|z|-N+1}.$

Therefore, for every ${N\geq1}$ and ${z\in\mathbb{C}}$,

$|\mathrm{e}_N(Nz)-\mathrm{e}^{Nz}\mathbf{1}_{|z|\leq1}| \leq \frac{N^N}{N!}\left(|z|^N\frac{N+1}{N+1-|Nz|}\mathbf{1}_{|z|\leq1} +|z|^{N-1}\frac{|Nz|}{|Nz|-N+1}\mathbf{1}_{|z|>1}\right).$

It remains to use the Stirling bound

$\sqrt{2\pi N}N^N\leq N!\mathrm{e}^N.$

Ginibre. Let us give an application to random matrices. Let ${M}$ be a random ${N\times N}$ complex matrix with independent and identically distributed Gaussian entries on ${\mathbb{C}}$ of mean ${0}$ and variance ${1/N}$ with density ${z\in\mathbb{C}\mapsto\pi^{-1}N\exp(-N|z|^2)}$. The variance scaling is chosen so that by the law of large numbers, asymptotically as ${N\rightarrow\infty}$, the rows and the columns of ${M}$ are stabilized: they have unit norm and are orthogonal in ${\mathbb{C}^N}$. The density of the random matrix ${M}$ is proportional to

$M\mapsto \prod_{1\leq j,k\leq N}\exp\Bigr(-N|M_{jk}|^2\Bigr) =\exp\left(-N\mathrm{Tr}(MM^*)\right).$

The spectral change of variables ${M=U(D+N)U^*}$, which is the Schur unitary decomposition, gives that the joint law of the eigenvalues of ${M}$ has density

$\varphi^{N,N}(z_1,\ldots,z_n) :=\frac{N^{\frac{N(N+1)}{2}}}{1!2!\cdots N!} \frac{\mathrm{e}^{-\sum_{i=1}^N N|z_i|^2}}{\pi^N} \prod_{i<j}|z_i-z_j|^2$

with respect to the Lebesgue measure on ${\mathbb{C}^N}$. This law is usually referred to as the “complex Ginibre Ensemble”. It is a well known fact that for every ${1\leq k\leq N}$, the ${k}$-th dimensional marginal distribution has density

$\begin{array}{rcl} \varphi^{k,N}(z_1,\ldots,z_k) &=&\int_{\mathbb{C}^{N-k}}\!\varphi^{N,N}(z_1,\ldots,z_N)\,\mathrm{d} z_{k+1}\cdots \mathrm{d} z_N \nonumber\\ &=&\frac{(N-k)!}{N!}\frac{\mathrm{e}^{-N(|z_1|^2+\cdots+|z_k|^2)}}{\pi^kN^{-k}} \det\left[(\mathrm{e}_N(Nz_i\overline{z}_j))_{1\leq i,j\leq k}\right], \end{array}$

where ${\mathrm{e}_N(w):=\sum_{\ell=0}^{N-1}w^\ell/\ell!}$ is the truncated exponential series.

We have seen above that uniformly on compact subsets of ${\{z\in\mathbb{C}:|z|\neq1\}}$,

$\lim_{N\rightarrow\infty}\varphi^{1,N}(z)=\frac{\mathbf{1}_{|z|\leq1}}{\pi}.$

Let us show now that

$\lim_{N\rightarrow\infty}(\varphi^{2,N}-(\varphi^{1,N})^{\otimes 2})=0$

uniformly on compact subsets of ${\{(z_1,z_2)\in\mathbb{C}^2:|z_1|\neq1,|z_2|\neq1,z_1\neq z_2\}}$.

Note that this cannot hold on ${\{(z,z):z\in\mathbb{C}, |z|\leq1\}}$ since ${\varphi^{2,N}(z,z)=0}$ for any ${N\geq2}$ and ${z\in\mathbb{C}}$ while ${\lim_{N\rightarrow\infty}\varphi^{1,N}(z)\varphi^{1,N}(z)=1/\pi^2\neq0}$ when ${|z|\leq1}$, and this phenomenon is due to the singularity of the interaction.

From the determinantal formula above ${\varphi^{1,N}}$ and ${\varphi^{2,N}}$ are given by

$\varphi^{1,N}(z)= \frac{\mathrm{e}^{-N|z|^2}}{\pi}\mathrm{e}_N(N|z|^2),\quad z\in\mathbb{C},$

where ${\mathrm{e}_N(w):=\sum_{\ell=0}^{N-1}w^\ell/\ell!}$ as usual, and, for every ${z_1,z_2\in\mathbb{C}}$,

$\begin{array}{rcl} \varphi^{2,N}(z_1,z_2) &=& \frac{N}{N-1} \frac{\mathrm{e}^{-N(|z_1|^2+|z_2|^2)}}{\pi^2} \big( \mathrm{e}_N(N|z_1|^2)\mathrm{e}_N(N|z_2|^2)-|\mathrm{e}_N(Nz_1\overline{z}_2)|^2 \big)\\ &=&\frac{N}{N-1}\varphi^{1,N}(z_1)\varphi^{1,N}(z_2) – \frac{N}{N-1} \frac{\mathrm{e}^{-N(|z_1|^2+|z_2|^2)}}{\pi^2} |\mathrm{e}_N(Nz_1\overline{z}_2)|^2. \end{array}$

It follows that for any ${N\geq2}$ and ${z_1,z_2\in\mathbb{C}}$,

$\begin{array}{rcl} \Delta_N(z_1,z_2) &=& \varphi^{2,N}(z_1,z_2)-\varphi^{1,N}(z_1)\varphi^{1,N}(z_2)\\ &=& \frac{1}{N-1}\varphi^{1,N}(z_1)\varphi^{1,N}(z_2) -\frac{N}{N-1}\frac{\mathrm{e}^{-N(|z_1|^2+|z_2|^2)}}{\pi^2}|\mathrm{e}_N(Nz_1\overline{z}_2)|^2. \end{array}$

In particular, using ${\varphi^{2,N}\geq0}$ for the lower bound,

$-\varphi^{1,N}(z_1)\varphi^{1,N}(z_2) \leq \Delta_N(z_1,z_2) \leq \frac{1}{N-1}\varphi^{1,N}(z_1)\varphi^{1,N}(z_2).$

From this and the error control for the exponential series above, we first deduce that for any compact subset ${K}$ of ${\{z\in\mathbb{C}:|z|>1\}}$

$\lim_{N\rightarrow\infty} \sup_{\substack{z_1\in\mathbb{C}\\z_2\in K}}|\Delta_N(z_1,z_2)| = \lim_{N\rightarrow\infty} \sup_{\substack{z_1\in K\\z_2\in\mathbb{C}}}|\Delta_N(z_1,z_2)| =0.$

It would remain to show that ${\lim_{N\rightarrow\infty}\Delta_N(z_1,z_2)=0}$ when in the same time ${|z_1|\leq1}$ and ${|z_2|\leq1}$. In this case ${|z_1\overline{z}_2|\leq1}$, and

$|\mathrm{e}_N(Nz_1\overline{z}_2)|^2 \leq 2\mathrm{e}^{2N\Re(z_1\overline{z}_2)} + 2r_N^2(z_1\overline{z}_2).$

Next, using the elementary identity

$2\Re(z_1\overline{z}_2)=|z_1|^2+|z_2|^2-|z_1-z_2|^2,$

we get

$\mathrm{e}^{-N(|z_1|^2+|z_2|^2)}| \mathrm{e}_N(Nz_1\overline{z}_2)|^2 \leq 2\mathrm{e}^{-N|z_1-z_2|^2} +2\mathrm{e}^{-N(|z_1|^2+|z_2|^2)}r_N^2(z_1\overline{z}_2).$

Since ${|z_1\overline{z}_2|\leq1}$, the formula for ${r_N}$ gives

$\mathrm{e}^{-N(|z_1|^2+|z_2|^2)}r^2_N(z_1\overline{z}_2) \leq \mathrm{e}^{-N(|z_1|^2+|z_2|^2-2-\log|z_1|^2-\log|z_2|^2)} \frac{(N+1)^2}{2\pi N}.$

Therefore, using the bounds ${\varphi^{1,N} \leq 1/\pi}$ and ${u-1 – \log u >0}$ for ${0<u<1}$, it follows that ${\lim_{N\rightarrow\infty}\Delta_N(z_1, z_2)=0}$ uniformly in ${z_1,z_2}$ on compact subsets of

$\{(z_1,z_2)\in\mathbb{C}^2:|z_1|<1,|z_2|<1,z_1\neq z_2\}.$

• The content of this post is mostly taken from preprint Dynamics of a planar Coulomb gas arXiv:1706.08776, by F. Bolley, J. Fontbona and myself.
• Random matrices, by M. L. Mehta (2004), chapter 15.
• Log-gases and random matrices, by P. Forrester (2010), chapter 15.
• L’Analyse au fil de l’histoire, by E. Hairer and G. Wanner (2001). This book in French is very pleasant and full of interesting historical details. Ernst Hairer is a well known expert in numerical analysis who turns out to be the father of the Fields medalist Martin Hairer.

Concerning the Taylor series of the exponential, I think the following result of Szegő (1924) is quite cute: Since $\mathrm{e}_N(z)$ is a polynomial of degree $N$ it has $N$ zeros. They have to escape from the complex plane as $N\to\infty$ because the exponential map has no zero there. It turns out they escape at speed $N$, and more precisely Szegő showed the $N$ zeros of the map $\mathrm{e}_N(Nz)$ accumulate on the curve $\gamma=\{z\in\mathbb C :\; |z\mathrm{e}^{1-z}|=1,\;|z|\leq 1 \}.$

2. Djalil Chafaï 2017-07-02

Very nice indeed, Adrien, thank you. By the way, this reminds me the paper entitled Zeros of sections of exponential sums by P. Bleher and R. Mallison, published in Int. Math. Res. Not. 2006, Art. ID 38937, 49 pp. DOI: 10.1155/IMRN/2006/38937. The historical reference seems to be G. Szegö, Über eine Eigenschaft der Exponentialreihe, Sitzungsber, Berliner Mathematische Gesellschaft 23 (1924), 500–564.

Apparently, this result by Gábor Szegö was reinvented about a decade later, independently and with another method, by Jean Dieudonné and published in Sur les zéros des polynômes-sections de $e^x$. Bull. Sci. Math., II. Ser. 59, 333-351 (1935). Below an embedded copy provided by Gallica. Note that this work of Dieudonné is cited in the paper by Bleher and Mallison mentioned above but with a wrong journal name!

3. Djalil Chafaï 2017-12-12

Yet another reference found later on:
Brian Conrey and Amit Ghosh
On the Zeros of the Taylor Polynomials Associated with the Exponential Function
The American Mathematical Monthly Vol. 95, No. 6 (Jun. – Jul., 1988), pp. 528-533.