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Libres pensées d'un mathématicien ordinaire Posts

Mathematics in Open Access : it works!

MathOAAn amazing news form the MathOA project:

The editors-in-chief and editorial board of Journal of Algebraic Combinatorics, currently published by Springer, have given notice of resignation. The Editors-in-Chief will see out their contracts with the old journal, until 31 December 2017. They will start a new journal, Algebraic Combinatorics, run according to the principles of Fair Open Access. The new journal, to be published in association with Centre Mersenne, is already open for business, with interim editors Victor Reiner and Satoshi Murai. To my knowledge this is the first time a mathematics journal has switched from a large commercial publisher to an open access model, let alone Fair Open Access. We hope it starts a trend, and congratulate the editors in chief Akihiro Munemasa, Hendrik van Maldeghem, Christos Athanasiadis and Hugh Thomas.

Centre Mersenne is a comprehensive publishing infrastructure offering modular services towards scientific communities and a dissemination platform for scientific publications, publishing in $\LaTeX$. It is developed by Mathdoc, in partnership with UGA Éditions, supported by CNRS and Université Grenoble Alpes (UGA), with funding from the Grenoble IDEX.

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Étymologies

Aléa. Du latin alea qui signifie « dé », « jeu de dés », « jeu de hasard ».

Hasard. De l’ancien français hasart, de l’espagnol azar, venant de l’arabe andalou الزهر, az-zahr, qui signifie « dé, jeu de dés », nommé d’après l’arabe زهر, zahr, qui signifie « fleur », car la face gagnante du dé portait une fleur.

Random. De l’ancien français randon qui signifie « course rapide ».

Stochastique. Du grec ancien στοχαστικός, stokhastikos, qui signifie « conjectural », de στόχος, stokhos, qui signifie « but, cible, conjecture ».

Source : wiktionnaire.

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About the exponential series

Leonhard Euler, the great master
Leonhard Euler, the great master

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.
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Inspiration exists but it has to find you working

Pablo Picasso dans son atelier - André Villers (1953)
Pablo Picasso dans son atelier – André Villers (1953)

Voici un petit quelque chose trouvé dans le manuscrit de l’habilitation à diriger des recherches que s’apprête à soutenir Omer Friedland :

Inspiration exists, but it has to find you working – Pablo Picasso.

En espagnol

La inspiración existe, pero tiene que encontrarte trabajando

in Las ciudades hablan: identidades y movimientos sociales en seis metrópolis latinoamericanas, Tomás R. Villasante (1994) p. 264.

Cette phrase de Picasso souligne à quel point les grands artistes sont avant tout de grands travailleurs. Cela fait penser aussi à la célèbre phrase d’Albert Einstein sur l’inspiration et la transpiration. Du reste, les scientifiques et les artistes partagent la même expérience de la créativité laborieuse et tumultueuse. N’est-ce pas aussi là que l’artisan devient artiste ? Il y a également un parallèle un peu plus distant avec les grands athlètes, dans le dépassement de soi et l’exploration sans fin des limites [insérer ici votre remarque sarcastique sur un footballeur].

De mon point de vue, il n’est pas responsable de laisser croire à la jeunesse qui peuple nos amphis que l’inspiration remplace l’effort, que la nature oppose le talent au labeur. Pour donner le meilleur de ses talents, il faut travailler inlassablement et rechercher l’élévation d’âme. Cela n’empêche pas d’assumer pleinement une certaine forme d’oisiveté choisie.

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