Nalini’s Lecture 2

Sequence of measured metric spaces satisfying the condition $\mathop{\mathrm{RCD}}(K,N)$ or $\mathop{\mathrm{CD}}(K,N)$. Here $K \leq$ Ricci curvature is a lower bound and dimension $\leq N$. The definitions are by Sturm, Lott-Villani and perhaps

(Gigli, Mondino, Savaré)

If $(X_{n},d_{n},\underbrace{x_{n}}_{\text{ Base Point }},m_{n}) \rightarrow^{\text{ GH }} (X_{\infty},d_{\infty},x_{\infty},m_{\infty})$ then the Cheeger (Dirichlet) energies are convergent in the “sense of Mosco” (convergence in the weak sense and in strong sense).

If $f_{n} \in L^{2}(m_{n}) \xrightarrow{\text{ Weak $L^2$ }}f_{\infty}$ then $\lim \mathcal{E}_{n}(f_{n},f_{n})\geq \mathcal{E}_{\infty}(f_{\infty},f_{\infty})$,

and for all $f \in L^{2}(m_{\infty})$, there is some $f_{n} \in L^{2}(m_{n})$ strongly converging to $f_{n} \in L^{2}(m_{n}) f$ and such that $\mathcal{E}(f_{n},f_{n}) \rightarrow \mathcal{E}_{\infty}(f,f)$.

(GMS)

If $X_{n},X_{\infty}$ are compact, then \[\begin{equation} \mathop{\mathrm{sp}}(\Delta_{X_{n}}) = \{ \lambda_{k}^{(n)} \rightarrow + \infty\}, n \in \mathbb{N} \cup \{\infty\}, \end{equation}\] or $\lambda_{k}^{(n)} \rightarrow \lambda_{k}^{(\infty)}$ with convergence of eigenfunctions and spectral projectors.

$(X_{n},d_{n},x_{n},m_{n}) \xhookrightarrow{\text{ isometry $i_{n}$}} (X,d)$ for $n \in \mathbb{N} \cup \{\infty\}$, then $f_{n} \rightarrow f_{\infty}$ in $L^{2}$ signifies that $\int_{}^{} \varphi f_{n} \circ i_{n}^{-1} \mathop{\mathrm{\,d}}i_{n} \sharp m_{n} \rightarrow \int_{}^{} f_{\infty} \circ i_{n} \mathop{\mathrm{\,d}}_{n} \sharp m_{\infty}$ for all $\varphi : X \rightarrow \mathbb{R}$ bounded and continuous. So $f_{n} \rightarrow_{L^{2}} f_{\infty}$ is more than just convergence of $L^{2}$ norms.

5. Homogenization

Take an open bounded set $\Omega \subseteq \mathbb{R}^{d}$. One has \[\begin{equation} L^{\mathcal{E}} = - \sum_{i,j}^{} \frac{\partial}{ \partial x_{i}} ( a_{ij}^{\varepsilon}(x) \frac{\partial}{\partial x_{j}}) + a_{0}^{ \varepsilon} \end{equation}\] One has Dirichlet conditions \[\begin{align} a_{ij}^{\varepsilon}(x) = & a_{ij}(\frac{x}{\varepsilon}) \\ a_{0}^{\varepsilon}(x) = & a_{0}(\frac{x}{\varepsilon}), \end{align}\] where $a_{ij},a_{0}$ are $\mathbb{Z}^{d}$-periodic, bounded. \[\begin{equation} a_{ij}(x) = \sum_{}^{} c_{k}(a_{ij}) e^{i k x }, \end{equation}\] where

\[\begin{equation} c_{0}(a_{ij}) = m(a_{ij}) \text{ average of }{a_{ij}}. \end{equation}\]

$L^{\varepsilon}$ converges to $L^{0}$ in the sense of the resolvent. Here \[\begin{equation} L^{0} = - \sum_{i,j}^{} \frac{\partial}{ \partial x_{i}} ( q_{ij} \frac{\partial}{\partial x_{j}}) + f \end{equation}\] with constant coefficients.

In 1 dimension, one has $L^{\varepsilon} : F \rightarrow (a^{\varepsilon} F' )'$. This converges in the sense of the resolvent to $L: f \rightarrow qf''$ where $q = \tfrac{1}{m(\frac{1}{a})} \neq m(a)$.

Note that for any $g$, $L^{\varepsilon}(f_{\varepsilon}) = (a^{\varepsilon} f_{\varepsilon}')' = g$ with Dirichlet conditions if whenever $f_{\varepsilon} \rightarrow f$ in the sense of $L^{2}$ then $f$ is a solution of $q f'' = g$.

6. Discrete tori

Consider $G_{N} = (\mathbb{Z}/N \mathbb{Z})^{2}$ or some other dimension. We consider the graph $G_{N}= (V_{N},E_{N})$. This graph is a discretization of the torus.

One has the discrete Laplacian diven as \[\begin{equation} \Delta_{N}^{\mathop{\mathrm{disc}}}: L^{2}(V_{N}) \rightarrow L^{2}(V_{N}) \end{equation}\] given by \[\begin{equation} \Delta_{N}^{\mathop{\mathrm{disc}}} f (x) = \sum_{y \sim x} [f(y) - f(x)] = \underbrace{{\sum_{y \sim x}{f(x)}}_{\text{ Adjacency matrix }}} - 4 f(x). \end{equation}\]

We have eigen functions \[\begin{equation} e_{k}(x) = e^{2 \pi i k\cdot x}. \end{equation}\] Then $x=(x_{1},x_{2}) \in ( \mathbb{Z}/N \mathbb{Z})^{2}$ , $k = (k_{1},k_{2}) \in ( \mathbb{Z}/ N \mathbb{Z})^{2}$. One has \[\begin{equation} \Delta_{N} e_{k} =- 4 [ \sin^{2} ( \frac{\prod_{}^{} k_{1}}{ N }) + \sin^{2} ( \frac{ \prod_{}^{} k_{2}}{ N}) ]. \end{equation}\]

1st limit

$N^{2} \Delta_{N}$ has eigenvalues $\Delta_{N} e_{k}$ converging to \[\begin{equation} 4 \pi^{2} k_{1}^{2}+ 4 \pi^{2} k_{2}^{2} \pi^2 k_2^2, \end{equation}\] with $k_{1},k_{2}$ fixed. Note that these are the eigenvalues of the torus $\mathbb{R}^{2}/\mathbb{Z}^{2}$.

There is a convergence in the sense of the resolvent generalized. If $z \in \mathbb{C} \setminus \mathbb{R}$, then \[\begin{equation} \| i_{N}(z - \Delta_{N}^{\mathop{\mathrm{disc}}}) i_{N}^{*} - i_{\infty} (z - \Delta_{\mathbb{R}^{2}}/\mathbb{Z}^{2}) i_{\infty}^{2} \| \rightarrow_{L^{2}} 0 \end{equation}\]

Here \[\begin{equation} \mathcal{H}_{N} = l^{2}(V_{N}), \mathcal{H}_{\infty} = L^{2} (\mathbb{T}), i_{\infty} = \text{ identity } \end{equation}\] and \[\begin{equation} i_{N}: \mathcal{H}_{N} \hookrightarrow \mathcal{H}_{0}, \end{equation}\] which is \[\begin{equation} f \mapsto \text{ Piecewise-constant function that makes $f(\frac{i}{N},\frac{j}{N})$ on a small neighbourhood}. \end{equation}\]

2nd limit

Convergence of “empirical spectral measures”. We have $|V_{N}| = N^{2}$ and $I$ a fixed interval. Then \[\begin{equation} \frac{1}{N^{2}} \mathop{\mathrm{\sharp}}{ (k_{1},k_{2}) , 4 ( \sin^{2}( \frac{ \prod_{}^{} k_{1}}{N}) + \sin^{2} ( \frac{\prod k_{2}}{N})) \in I }. \end{equation}\] This is a Riemann sum approximating the integral \[\begin{equation} \int_{[0,1]^{2}}^{} \mathbb{I}_{I}(4 \sin^{2} ( \pi x_{1}) + 4 \sin^{2}(\pi x_{2}))\mathop{\mathrm{\,d}}(x_{1},x_{2}) = \langle \delta_{0}, P_{I}( \Delta_{\mathbb{Z}^{2}})\rangle_{l^{2}(\mathbb{Z}^{2}) \delta_{0}}, \end{equation}\] where $\delta_{0}$ is the Dirac mass at $0 \in \mathbb{Z}^{2}$.

This means that $(\mathbb{Z}/N \mathbb{Z})^{2}$ converges towards $\mathbb{Z}^{2}$ in the sense of Benjamini-Schramm.

Convergence in the sense of Benjamini-Schramm

For discrete graphs

Let $\mathbb{G}^{d}_{*} = \{ \text{ graphs of valency} \leq d, \text{ rooted }\}/ \sim$ where $\sim$ is equivalence upto root-preserving isomorphisms. We equip this with a distance. \[\begin{equation} \mathop{\mathrm{dist}}((G,x),(G',x')) \leq \varepsilon \text{ if there is an isomorphism } B_{G}(x, \tfrac{1}{\varepsilon}) \sim B_{G'}( x', \frac{1}{\varepsilon}). \end{equation}\]

In fact $\mathbb{G}^{d}_{*}$ is compact with respect to this metric. We denote \[\begin{equation} \mathcal{P}^{1}(\mathbb{G}^{d}_{*}) = \{ \text{Probability measures on }\mathbb{G}^{d}_{*}\}. \end{equation}\] We equip $\mathcal{P}^{1}(\mathbb{G}^{d}_{*})$ with the weak topology.

For a finite graph $G=(V,E)$ , $\gamma_{G} \in \mathcal{P}^{1}(\mathbb{G}_{*}^{d})$ where \[\begin{equation} \gamma_{G} = \frac{1}{|V|} \sum_{x \in V} \delta_{(G,x)} \end{equation}\]

Let $G_{N} = (V_{N},E_{N})$ be a sequence of finite graphs. We say that $G_{N}$ converges in the sense of BS (= locally weak convergence) if $\gamma_{G_{N}}$ has a weak limit $\mathbb{P}_{\infty} \in \mathcal{P}^{1}(\mathbb{G}^{d}_{*})$.

If $(G_{N}) \xrightarrow{BS} \mathbb{P}_{\infty}$. Let $(\lambda_{k}^{(N)})^{N}$ be the eigenvalues of the adjacency matrix $A_{g}$ of $G_{N}$, so they lie in $[-d,d]$. Then for $f \in C^{\circ} ( \mathbb{R})$ one has \[\begin{equation} \frac{1}{|V_{N}|} \sum_{k=1}^{N} f(\lambda_{k}^{(N)}) \xrightarrow{N \rightarrow + \infty} {\underbrace{\mathbb{E}_{\infty}}_{\text{ with respect to the measure $\mathbb{P}_{\infty}$ }}}( \langle \delta_{0}, f(A_{g}) \delta_{0}\rangle_{l^{2}(g)}). \end{equation}\]

One has $A_{g}$ is the adjacency matrix of the random graph $g$ and $\delta_{0}$ = Dirac mass at 0.

Note that this implies \[\begin{equation} \text{ for all } (H,y) \in \mathbb{G}_{*}^{d} \text{ for all } R \geq 0 \text{ fixed }, \frac{1}{ |V_{N}|} \mathop{\mathrm{\sharp}}\{ x \in V_{N} , B_{G_{N}(x,R) \sim B_{H}(y,R)}\} \end{equation}\] has a limit $\mathbb{P}_{\infty}( B_{g}(0,R) \sim B_{H}(y,R))$.

Example of discrete torus

Suppose $R$ is given and $x \in (\frac{\mathbb{Z}}{ N \mathbb{Z}})^{2}$ for $N \gg R$. Let $B_{G_{N}} ( x, R) \sim B_{\mathbb{Z}^{2}} (0,R)$
and $\mathbb{P}_{\infty} = \delta_{(\mathbb{Z}^{2},0)}$, the limit graph is deterministic!

For the discrete graph $[[0,N-1]]^{2}$ (graph does no wrap around) , if $x = (a,b) \in [[R,N-R]]$, then $B_{G_{N}}(x,R) \sim B_{\mathbb{Z}^{2}}(0,R)$ but not if $a$ or $b$ is outside $[[R,N-R]]$. The limit is still $\delta_{(\mathbb{Z}^{2},9)}$.

(of the theorem of BS)

The $\lambda_{k}^{(N)} \in [-d,d]$. It suffices to test $f(\lambda) = \lambda^{R}$ for an integer $ R $. Then \[\begin{equation} \frac{1}{|V_{N}|} \sum_{k \geq 1}^{|V_{N}|} (\lambda_{k}^{(N)})^{R} = \frac{1}{|V_{N}|} \mathop{\mathrm{Tr}}(A^{R}_{G_{N}}) = \frac{1}{|V_{N}|} \sum_{x \in V_{N}}^{} A^{R}_{G_{N}}(x,x) \end{equation}\] Take $\varphi(G,x) = A_{G}^{R}(x,x)$. This is continuous on $\mathbb{G}_{*}^{d}$ since the number of paths of length $R$ from $x$ to itself do not depend on anything but $B_{G}(x,R)$. So the original equation converges to \[\begin{equation} \mathbb{E}_{\infty}( \varphi (f,0)) = \mathbb{E}_{\infty}( \langle \delta_{0}, A_{g}^{R} \delta_{0}\rangle) \end{equation}\]

Lecture 2

4. Sequence of measured metric spaces