The vase of Soissons

% Fragments with mass $\approx2^x\cdot10$ g

Homogeneity / self-similarity

Homogeneous:   all fragments split at the same rate.

Self-similar:   fragments with size $x$ split at rate $\propto x^\alpha$.

(Filippov, 1961; Bertoin, 2003) For $\alpha<0$, the fragments vanish in finite time.
For $\alpha>0$, the fragments exhibit a non-Gaussian distribution.

What is a growth-fragmentation?

The fragments' genealogy

Asymptotic distribution $\,\tiny(\alpha=0)$

(D., 2017) Let $\{\!\!\{\class{red}{X_i}\}\!\!\}$ be a homogeneous growth-fragmentation and $$h(q)=\frac1q\log E\!\left[\sum_{i=1}^\infty\class{red}{X_i}(1)^q\right]\!.$$ Suppose $h'(q)<0$. Then $$e^{-qh(q)t}\sum_{i=1}^\infty\class{red}{X_i}(t)^q\,\delta_{\log\class{red}{X_i}(t)}\underset{t\to\infty}\approx M_\infty(q)\cdot\mathcal N(t\mu_q,t\sigma^2_q).$$

The largest fragment $\,\tiny(\alpha=0)$

(D., 2017) Let $\class{red}{Y}(t)$ be the mass of the largest cell at time $t$. Then $$\log\class{red}{Y}(t)\underset{t\to\infty}\approx h(\bar q)t-\frac3{2\bar q}\log t+G,$$ where $\bar q=\operatorname{argmin}h$ and $G$ is some random variable.

Asymptotic distribution $\,\tiny(\alpha>0)$

(D., 2017) Let $\{\!\!\{\class{blue}{X_i}\}\!\!\}$ be a self-similar growth-fragmentation with $\alpha>0$.
Suppose Cramér's hypothesis holds:
$\displaystyle q\mapsto\log E\!\left[\sum_{i=1}^\infty\class{red}{X_i}(1)^q\right]\!$ looks like . Then, for some (non-Gaussian) probability distribution $\rho$, $$\sum_{i=1}^\infty\class{blue}{X_i}(t)^{\omega}\,\delta_{t^{1/\alpha}\class{blue}{X_i}(t)}\underset{t\to\infty}\longrightarrow M_\infty(\omega)\cdot\rho.$$

The largest fragment $\,\tiny(\alpha>0)$

(D., 2017) Let $\class{blue}{Y}(t)$ be the mass of the largest cell at time $t$.
Then (under Cramér's hypothesis) $$\log\class{blue}{Y}(t)\underset{t\to\infty}\approx -\frac1\alpha\log t.$$

Homogeneity / self-similarity

From mass $x$, $\class{blue}{X_\alpha}$ moves $x^\alpha$ times faster than $\class{red}{X_0}$.

From mass $x$, moves $x^\alpha$ times faster than .

That is, if
$(t)\approx x+t\cdot s$, then $(t)\approx x+t\cdot\class{blue}{x^\alpha}s$. Thus   $(t)={}$$\Bigl(\int_0^t$$\class{blue}{(s)^\alpha}\,\mathrm{d}s\Bigr)$.

Discrete fragmentation tree

$\class{blue}{\mathcal T^{(11)}}$

Scaling limit

(Haas & Miermont, 2012) Suppose $n\overset{\smash{\tiny\textsf{frag.}}}\longmapsto\{np_1\ge\cdots\ge np_r\}$ has probability $q_n({\rm d}{\bf p})$ and "macroscopic" dislocations are rare, i.e., $$n^{-\alpha}\cdot (1-p_1)\,q_n({\rm d}{\bf p})\underset{n\to\infty}\longrightarrow(1-p_1)\,\nu({\rm d}{\bf p}),$$ for some $\alpha<0$ and dislocation measure $\nu$. Then $$n^\alpha\cdot\mathcal T^{(n)}\underset{n\to\infty}\longrightarrow\mathcal T,$$ the genealogical tree of a self-similar pure fragmentation in which $x\overset{\smash{\tiny\textsf{frag.}}}\longmapsto\{xp_i\}_{i\ge1}$ occurs at rate $x^\alpha\cdot\nu({\rm d}{\bf p})$.

Discrete growth-fragmentation

Call $\class{blue}{\mathcal T^{(8,2)}}$ this tree and $\class{red}{{\bf X}^{(8,2)}}(k)$ the molecules at time $k$.

Scaling limit

(D., 2019+) Suppose $n\overset{\smash{\tiny\textsf{growth}}}\longmapsto k$ or $n\overset{\smash{\tiny\textsf{frag.}}}\longmapsto\{k,n-k\}$ with probability $p_{n,k}$. Under assumptions on $n^{-\alpha}\cdot p_{n,k}$ as $n\to\infty$, we can fix $M$ so that
$\displaystyle\frac1n{\bf X}^{(n,M)}\!\left(\lfloor n^{-\alpha}\cdot\rfloor\right)\!\quad$and$\displaystyle\quad n^\alpha\cdot\mathcal T^{(n,M)}$ respectively approach a self-similar growth-fragmentation with index $\alpha<0$ and its associated genealogical tree.