Length statistics of random multicurves on closed hyperbolic surfaces (Q2102154)

Let \(X = (S, \rho)\) be a closed oriented connected hyperbolic surface of genus at least 2. A finite ordered \(k\)-tuple \(\gamma = (m_1 \gamma_1,\dots,m_k \gamma_{k})\) is called \textit{an ordered multi-geodesic} on \(X\) if for every \(i\), \(m_i \geq 1\), \(\gamma_i\) is a simple closed geodesic on \(X\) and for \(i \neq j\), \(\gamma_i \cap \gamma_j = \emptyset\). Denote the mapping class group of \(S\) by \(\mathrm{Mod}(S)\) and the standard \((k-1)\)-simplex by \[ \Delta^{k-1} = \{(x_1,\dots,x_k) \in \mathbb{R}_{\geq 0}^{k}: \sum_ix_i=1\}. \] Fix an ordered multi-geodesic \(\gamma\) and let \(s_{X,R,\gamma}\) be the (finite) set of ordered multi-geodesics of the same type as that of \(\gamma\) with length at most \(R\). That is, \[ s_{X,R,\gamma} = \{\alpha \in \mathrm{Mod}(S)\cdot\gamma: \ell_{X}(\alpha) \leq R\}, \] where \[ \ell_{X}(\alpha) = \sum_i m_i\ell_{X}(\alpha_i) \] and \(\ell_X(\alpha_i)\) is the \(X\)-length of the unique geodesic representative of \(\alpha_i\). Each finite set \(s_{X,R,\gamma}\) gives rise to a finite subset \(\widehat{\ell}_{X,R,\gamma}(s_{X,R,\gamma})\) of \(\Delta^{k-1}\) via the length map \[ \widehat{\ell}_{X,R,\gamma}: s_{X,R,\gamma} \to \Delta^{k-1}, \ (m_1\alpha_1,\dots,m_k\alpha_k) \mapsto \frac{1}{\ell_X(\alpha)}(m_1\ell_X(\alpha_1),\dots,m_k\ell_X(\alpha_k)). \] For each \(R > 0\), consider the probability measure \(\mu_{X,R,\gamma}\) on \(\Delta^{k-1}\) given by the pushforward of the uniform measure on \(s_{X,R,\gamma}\), namely, \[ \mu_{X,R,\gamma} = \left(\widehat{\ell}_{X,R,\gamma}\right)_{*}\left(\frac{1}{|s_{X,R,\gamma}|}\sum_{\alpha \in s_{X,R,\gamma}} \delta_{\alpha}\right) \] where \(\delta_{\alpha}\) is the Dirac measure at \(\alpha \in \mathrm{Mod}(S)\cdot \gamma\). The paper under review addresses the following equidistribution problem. Fix \(\gamma\) and \(X\), what is the limit measure on \(\Delta^{k-1}\) of \(\mu_{X,R,\gamma}\) as \(R \to \infty\)? When \(k = 3g-3\), this was answered by \textit{M. Mirzakhani} in [``Counting mapping class group orbits on hyperbolic surfaces'', Preprint, \url{arXiv:1601.03342}]. Following Mirzakhani's strategy, the author answers the above equidistribution theorem for general \(k\). Theorem 1.2. Let \(\gamma = (m_1\gamma_1,\dots,m_k\gamma_k)\) be an ordered multi-geodesic and \(X\) be a hyperbolic surface. Then there is a homogeneous polynomial \(\overline{P}_{\gamma}\) in \(k\) variables associated to \(\gamma\) such that, up to a constant, \(\mu_{X,R,\gamma}\) converges weakly to the measure \(\overline{P}_{\gamma}(x_1/m_1,\dots,x_k/m_k) \lambda(x)\) on \(\Delta^{k-1}\) as \(R\) goes to infinity, where \(\lambda(x)\) is the Lebesgue measure on \(\Delta^{k-1}\). Note that the polynomial \(\overline{P}_{\gamma}\) depends only on \(\gamma\) and is independent of the hyperbolic structure \(X\). See also [\textit{F. Arana-Herrera}, Geom. Dedicata 210, 65--102 (2021; Zbl 1461.30104)] for related results.