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
\[ \{(z_1,z_2)\in\mathbb{C}^2:|z_1|<1,|z_2|<1,z_1\neq z_2\}. \]
Further reading.
- 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.