I am attending Nalini’s College de France course. This is Lecture 4.
Let \((X_{n},d_{n},m_{n})\) be a sequence of metric spaces such that \(m_{n}(X_{n}) < \infty\) and it satisfies the condition \(\mathop{\mathrm{RCD}}(K,N)\) for Ricci curvature \(\geq K\) and dimension \(\leq N\). We assume that \((X_{n},d_{n},m_{n}) \rightarrow _{BS} (X_{\infty}, d_{\infty},x,m_{\infty})\) random in \(\mathbb{P}_{\infty}\) and \(m_{n}(B(x,1)) \geq v_{0} > 0\) uniformly for all \(x,n\). Then for all \(t > 0\), we have \[\begin{equation} \frac{1}{m_{n}(X_{n})}\mathop{\mathrm{Tr}}(e^{t \Delta_{X_{n}}}) \rightarrow _{n \rightarrow \infty} \mathbb{E}_{\infty}(e^{t \Delta x_{\infty}} ( x,x)), \end{equation}\] which is \[\begin{equation} = \frac{1}{m_{n}(X_{n})} \int_{X_{n}}^{} p_{t}^{X_{n}}\mathop{\mathrm{\,d}}m_{n}(z) = \frac{1}{m_{n}(X_{n})} \sum_{j=1}^{+ \infty} e^{-t \lambda_{j}^{(n)}}. \end{equation}\]
The proof uses results of Gigle-Mondino-Savaré on the stability of \(p_{t}^{X_{n}}\), along with a priori boundedness of \(p_{t}\) which demands \(m(B(x,1)) \geq v_{0}\).
1) Let \(X_{n} = \Gamma_{n} \backslash G / K\) be a compact set under the hypothesis that \(X_{n} \rightarrow_{\text{ BS }} G / K\) and that the injectivity radius is bounded from below (the base point is not relevant here (one can take it as the origin)).
2) Let \(\tilde{M} =\) universal cover of a manifold \(M\), which is a compact Riemannian manifold. Let \(\Gamma_{n} \subseteq \mathop{\mathrm{Iso}}(\tilde{M})\) such that \(X_{n} = \Gamma_{n} \backslash \tilde{M}\) is compact. Then, if the injectivity radius \(\rightarrow + \infty\) then \(X_{n} \rightarrow \frac{1}{\mathop{\mathrm{vol}}(D)} \int_{D}^{} \delta_{(\tilde{M},x)}\mathop{\mathrm{\,d}}\mathop{\mathrm{vol}}(x)\), where \(D \subseteq \tilde{M}\) is a fundamental domain of \(M= \Gamma \backslash \tilde{M}\).
3) Compact random hyperbolic surfaces also are convergent in the sense of Benjamini Schramm.
In the case of Example 1), on has for each \(k\) and \(\Delta_{(k)}\) being the Laplacian on \(k\)-forms that \[\begin{equation} \frac{1}{\mathop{\mathrm{vol}}(X_{n})} \mathop{\mathrm{Tr}}(e^{t \Delta_{(k)}^{X_{n}}}) \rightarrow_{n \rightarrow \infty} \mathop{\mathrm{Tr}}( e^{t \Delta^{X_{n}}_{(k),L^{2}}}) (0,0). \end{equation}\] Here \(\Delta_{(k),L^{2}}\) is the Laplacian on \(L^{2}\) \(k\)-forms. When \(t=\infty\), this implies \[\begin{equation} \frac{f_{k}(X_{n},\mathbb{R})}{\mathop{\mathrm{vol}}(X_{n})} \rightarrow ?? \end{equation}\]
(Elek, Bowen, Albert-Bergeron,Beiringet, Gelander)
The \((X_{n},d_{n},m_{n}) \rightarrow _{\text{ BS }} (X_{\infty},d_{\infty},m_{\infty})\) that satisfies “good conditions” then for all \(k\) one has \(\frac{f_{k}(X_{n})}{m_{n}(X_{n})}\) has a limit when \(n \rightarrow \infty\) defined with simplicial complexes.
In the case of random regular graphs (according to Friedman), one has the following model. It’s called ofen the “algebraic model”.
Here we take \(N\) vertices and \(d/2\) edges per vertex. That is \(Nd\) pairs of vertices and edges. This means the number of graphs is \((Nd-1)(Nd-3) \dots 5 \cdot 3 \cdot 1 = (Nd-1)!!\) (how?).
For a ranadom graph, one has the eigenvalues \[\begin{equation} \lambda_{0} > \lambda_{1} \geq \dots \geq \lambda_{N-1} . \end{equation}\] Here the multiplicity of \(\lambda_{0}\) is the number of connected components. We then take \[\begin{equation} \frac{1}{N} \sum_{i=0}^{N-1} f(\lambda_{i}) \rightarrow_{N \rightarrow + \infty} \langle \delta_{0}, f( A_{t_{d}}) \delta_{0} \rangle _{l^{2}(T_{d})} = \int_{}^{} f(\lambda)\mathop{\mathrm{\,d}}m(\lambda), \end{equation}\] where \[\begin{equation} dm(\lambda) = \frac{(q+1) \sqrt{4 q - \lambda^{2}}}{ 2 \pi ( (q+1)^{2} - \lambda^{2})} \mathbb{I}_{[-2 \sqrt{q}, 2 \sqrt{ q}]}(\lambda) \mathop{\mathrm{\,d}}\lambda \end{equation}\]
This in particular shows the conjecture of Alon by Joel Friedman in 2008. That is for a random graph \(G_{N}\) one has \[\begin{equation} \mathbb{P}_{N}( \mathop{\mathrm{sp}}(A_{G_{N}}) \setminus \lambda_{0} \subseteq [- 2 \sqrt{q} - \varepsilon, 2 \sqrt{q} + \varepsilon] ) \rightarrow _{N \rightarrow + \infty} 1. \end{equation}\]
For a regular graph \(G = (V,E)\), the matrix of Hashimoto \(B\) is defined by
Then, we observe that \[\begin{equation} \mathop{\mathrm{Tr}}( \frac{B^{k}}{q^{k/2}}) = \sum_{\substack{\gamma \text{ oriented geodesic } \\ \mathrm{len}(\gamma) = k}}^{} T(\gamma), \end{equation}\] where \(T(\gamma)\) is the smallest period of \(\gamma\). The last expression is \[\begin{equation} = \sum_{j=0}^{N-1} q^{\pm i k s_{j}} + \frac{N(d-2)}{2} \frac{1 + (-1)^{k}}{q^{\frac{k}{2}}}. \end{equation}\]
If \(\varphi: \mathbb{Z} \rightarrow \mathbb{C}\) is an even function, then \[\begin{equation} \mathop{\mathrm{Tr}}( \sum_{l >0}^{} \varphi(l) \frac{B^{l}}{q^{l/2}}) = \sum_{j=0}^{N-1} \widehat{\varphi}(s_{j}) - N \varphi(0) + \frac{N (d-2)}{2} \sum_{l > 0}^{} \varphi(l) \frac{1 + (-1)^{l}}{ q^{l/2}}, \end{equation}\] where \(\varphi\) is of bounded support in \([-L,L]\) and \[\begin{equation} \widehat{\varphi}(s) = \sum_{l \in \mathbb{Z}}^{} \varphi(l) q^{i l s} \end{equation}\] is the Fourier-transform of period \(\tfrac{2 \pi }{ \log q}\). For finitely supported \(\varphi\), this is indeed a trignometric polynomial \(P_{L}(\lambda)\) whose degree is controlled by \(L\).
Observe that \(q^{ils} + q^{-ils} = P_{l}(q^{is} + q^{-is})\) (what?).
One has \[\begin{equation} \frac{1}{N} \widehat{\varphi}(s_{j}) = \int_{ \mathbb{R} / ( \frac{2 \pi }{ \log q} \mathbb{Z} )}^{} \widehat{\varphi}(s) \mathop{\mathrm{\,d}}m(s) + \frac{1}{N} \sum_{ \gamma \text{ oriented geod. }}^{} \frac{T(\gamma)}{q^{\frac{l(\gamma)}{2}}} \varphi(l(\gamma)). \end{equation}\] The measure of \(m\) is defined by the Fourier coefficients.
We have \(l(\gamma)\) = period of \(\gamma\) and \(T(\gamma)\) = primitive period. So \(l(\gamma) = n T(\gamma)\) where \(n \geq 1\). One has \(\int_{}^{} \mathop{\mathrm{\,d}}m\) and \(\int_{}^{} q^{i s l }\mathop{\mathrm{\,d}}m = - \frac{d-2}{2} \frac{(1+ (-1)^{l})}{q^{l/2}}\). Here \(m\) is the same measure comes from the change of variables \(\lambda = q^{\frac{1}{2} + i s } + q^{\frac{1}{2} - i s}.\)
For \(\Gamma = (V(\Gamma), E(\Gamma))\) is a graph of vertices and edges whose degree is \(\leq d = q+1\). (actually we had \(q = d-1\) all this time,)
Consider the labelings \[\begin{equation} x : V(\Gamma) \rightarrow [[1,\dots,N]], \end{equation}\] \(t\) be labels in \([[1,\dots,d]]\) on the \(1/2\)-edges of Gamma.
Denote \[\begin{equation} \mathbb{P}_{N} ((\Gamma,x,t) \text{ is realized in the configuration model}) = \frac{1}{(Nd-1)!!} (Nd-2a-1)(Nd-2a - 3) \dots 5 \cdot 3 \cdot 1 = \frac{(Nd-2a-1)!!}{(Nd-1)!!}. \end{equation}\]
Then by definition \[\begin{equation} \mathbb{E}_{N}( \text{ number of realizations of $\Gamma$ in $G_{N}$ }) = \frac{(Nd - 2a -1)!!}{ (Nd-1)!!} \frac{\mathop{\mathrm{\sharp}}\{(x,t) \text{ possible }\}}{ N (N-1) \cdot (N-v+1)}, \end{equation}\] and we have \[\begin{equation} \prod_{i=1}^{m} (d(d-1))^{l_{i}-1} \times C(F) , \end{equation}\] where \(C(F)\) are the labelings of \(1/2\)-edges that take share the vertices of \(F\).
This was followed by some calculations that I did not follow. But the point seems to be that \(\mathbb{E}_{N}\) is a rational polynomial in \(\tfrac{1}{N}\) and that makes all the difference.
For \(f_{a}(x) = (1-x)(1-2x) \dots (1- ax)\) for \(a \in \mathbb{N}\), we write this as \(1 - x Q_{1}(a) + x^{2} Q_{2}(a) + \dots + x^{n+1} F_{n+1}(x,a)\).
We have \(g_{n}(x) = \frac{1}{ (1-x)(1-2x) \dots (1-ax) } = 1 + x R_{1}(a) + \dots x^{n+{1}} G_{n+1}(x,a)\).
The terms not mentioned are small in a quanititative sense. Quantifying this is important in the polynomial method.
More next time.
This page was updated on November 28, 2025.
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