Sharp asymptotics for Toeplitz determinants and convergence towards the Gaussian free field on Riemann surfaces

Theorem 1.1. Let X be a genus g Riemann surface and consider the canonical determinantal point process on X with N particles. It satisfies the following Moser-Trudinger type inequality: $$(1.10) \qquad \log \mathbb{E}(e^{-(\sum_{i=1}^{N} (\phi(x_i) - \int_X \phi \frac{\omega}{V}))}) \le \left(\frac{1}{1 + (1-q)/N} + \epsilon_N\right) \frac{1}{2} \|d\phi\|_X^2 + \epsilon_N$$ where the error term $\epsilon_N$ is exponentially small, i.e., $\epsilon_N \leq Ce^{-N\delta}$ for some postive number C and $\delta$ independent of $\phi$ and where $\delta$ can be explicitly expressed in terms of the injectivity radius of $(X,\omega)$ (see formula 2.4 in Prop 2.1). Similarly, (1.11) $$\log \mathbb{E}(e^{-(\sum_{i=1}^{N} (\phi(x_i) - \mathbb{E}(\phi(x_i)))}) \le \left(\frac{1}{1 + (1 - g)/N} + \epsilon_N\right) \frac{1}{2} \|d\phi\|_X^2 + \epsilon_N \|\phi\|_{L^1(X)/\mathbb{R}} + \epsilon_N$$ Moreover, when X is the Riemann sphere (i.e. g = 0) all the error terms above vanish identically. An important ingredient in the previous proof is a convexity result of Berndtsson [8] which in this particular case essentially amounts to the positivity of a certain determinant line bundle over the space of all Kähler metrics in the first Chern class of L. The error terms $\epsilon_N$ above come from the error terms in the Yau-Tian-Zelditch-Catlin expansion [44, 2, 33, 32] for the underlying Bergman kernel. As follows from Theorem 3.1 below these error terms are exponentially small, slightly refining previous recent results in [31, 32] (see section 3 for precise formulations). As a simple consequence of the previous theorem we then obtain a sharp version of the tail estimate 1.8 for such canonical processes. The main point is that it shows that the error term o(1) appearing in the estimate 1.8 can be taken to be independent of the function $\phi$ . As a consequence the estimate holds with minimal regularity assumptions on $\phi$ :

Corollary 1.2. Let X be a genus g Riemann surface and consider the canonical determinantal point process on X with N particles. Let $\phi$ be a function on X such that its differential $d\phi$ is in $L^2(X)$ . Then the linear statistic defined by $\phi$ has an exponentially decaying tail: $$\epsilon_{N,\lambda}(\phi) \le 2\exp\left(-N^2\left(\frac{2\lambda^2}{\|d\phi\|_X^2\left(1+\frac{(1-g)}{N}\right)+\epsilon_N\right)}+\epsilon_N\right)\right)$$ where the error terms $\epsilon_N$ are as in the previous theorem. We will also show that the Moser-Trudinger inequality in Theorem 1.1 is in fact an asymptotic equality in the following sense:

Theorem 1.3. (strong Szegö type theorem). Let X be a genus g Riemann surface and consider the canonical determinantal point process on X with N particles. Let $\phi$ be a complex valued function on X such that its differential is in $L^2(X,\mathbb{C})$ , i.e. $\phi$ has finite Dirichlet norm. Then $$\log \mathbb{E}(e^{-(\sum_{i=1}^{N} (\phi(x_i) - \int_X \phi\omega))}) \to \frac{1}{2} \int_X d\phi \wedge d^c \phi$$ as $N \to \infty$ and the same convergence holds when the exponent above is replaced with the fluctuation of the linear statistic of $\phi$ . In [3] it was shown that, as long as $\omega > 0$ and $\phi$ is smooth an analogue of the convergence above holds in any dimension n if the conformally invariant norm above is replaced by the Dirichlet norm wrt $\omega$ . But it should be emphasized that when n > 1 the convergence does not hold if one relaxes the smoothness assumption on $\phi$ to allowing a gradient in $L^2$ (see section 2.4 for counter examples). The previous theorem may be equivalently formulated as the following Central Limit Theorem (CLT), valid under minimal regularity assumptions:

Corollary 1.4. (CLT) The fluctuations $\delta_N - \mathbb{E}(\delta_N)$ of the empirical measure $\delta_N$ converge in distribution to the Laplacian (or rather $dd^c$ ) of the Gaussian free field (GFF). In other words, for any $\phi \in L^1(X)$ with $d\phi \in L^2(X)$ the fluctuations $$\sum_{i=1}^{N} (\phi(x_i) - \mathbb{E}(\phi(x_i)))$$ of the corresponding linear statistics converge in distribution to a centered normal random variable with variance $\|d\phi\|_X^2$ . The GFF is also called the massless bosonic free field in the physics litterature. Heuristically, this is a random function wrt the Gaussian measure on the Hilbert space of all $\phi$ (mod $\mathbb{R}$ ) equipped with the Dirichlet norm $\|d\phi\|_X^2/2$ . For the precise definition of the GFF and its Laplacian see [40] (Prop 2.13 and Remark 2.14) and for a comparison with the physics litterature on Coulomb gases see section 1.3 in [41].

Proposition 2.1. Let $L \to X$ be a line bundle of degree V over a Riemann surface of genus g. Assume that L is equipped with a metric $e^{-\Phi}$ with strictly positive curvature form $\omega(=dd^c\Phi)$ such that the Riemannian metric on X defined by $\omega$ has constant scalar curvature R(=(2-2g/V). Then the canonical determinantal point processes associated to kL (with $N(=N_k)$ particles) satisfy $$\sup_{X} \left| \frac{\mathbb{E}_{N}(\delta_{N}/N)}{\omega/V} - 1 \right| \le \epsilon_{N},$$ where $\epsilon_N$ is exponentially small, i.e. $\epsilon_N \leq Ce^{-\delta N}$ . In the case g=0 we have $\epsilon_N=0$ and when g>0 the constant $\delta$ can be taken to be arbitrarily close to (2.4) $$\frac{2}{RV}\log(\cosh(\sqrt{\frac{\pi R}{2}}I(X)))$$ where I(X) is the injectivity radius of X (which coincides with half the length of the shortest geodesic on X). Proof. To simplify the notation we set V=1 (the case $V\neq 1$ follows from trivial scalings). First we recall that in the general setting where 1.4 defines a Hilbert norm on $H^0(X,kL)$ we have the basic relation $\mathbb{E}_N(\delta_N)=B_k(x)d\nu$ where $B_k(x)$ is the point-wise norm of the corresponding Bergman kernel and hence Theorem 3.1 below (or its corollary) gives $\mathbb{E}_N(\delta_N)=k+R/2+\mathcal{O}(e^{-\delta k})$ . Since, $N=\int B_k(x)d\nu$ it follows that N=k+R/2 for k>>1 (a special case of the Riemann-Roch theorem) concluding the proof of the proposition.

Proposition 2.2. Let $L \to X$ be a line bundle over a Riemann surface equipped with a smooth metric with strictly positive curvature form $\omega$ . Consider the corresponding adjoint determinantal point process. Then the following estimate holds $$\frac{1}{N}\log \mathbb{E}(e^{-\phi}) - \mathcal{E}_{\omega}(\phi) \le \sup_{X} \left| \frac{\mathbb{E}(\delta/N)}{\omega/V} - 1 \right| \left( -\mathcal{E}_{\omega}(\phi - \sup_{X} \phi) \right)$$ for any smooth function $\phi$ satisfying $\omega_{\phi} := dd^c \phi + \omega \geq 0$ , where $\delta(=\delta_N)$ denotes the empirical measure of the process with N particles. The proof is a simple modification of the proof Theorem 33 in [5]. As a courtesy to the reader we will recall the argument in [5]. An important ingredient in the proof is the notion of a $C^0$ -geodesic (wrt the Mabuchi metric) connecting $\phi_0$ and $\phi_1$ in $C^0(X) \cap PSH(X, \omega)$ . This may be defined as the continuous path $\phi_t (= \phi(\cdot,t))$ connecting $\phi_0$ and $\phi_1$ in $C^0(X) \cap PSH(X, \omega)$ obtained as the upper envelope of all $S^1$ - invariant $\pi^*\omega$ -psh extensions to the n+1-dimensional complex manifold with boundary $M := X \times ([0,1] \times S^1)$ (where $\pi$ denotes the projection from $X \times [0,1[\times S^1$ to X)). In particular, $\phi_t$ is convex in the real parameter $t \in [0,1]$ and satisfies the homogenous Monge-Ampère equation in the interiour of M: (2.5) $$\partial_t \partial_t \phi_t - |(\bar{\partial}_X (\partial_t \phi_t))|^2_{\omega_{\phi_t}} = 0$$ in the weak sense of pluripotential theory (see [5] for the precise construction). The following variational formulae are well-known (and straight-forward): (2.6) $$(i) - \frac{1}{N} d(\log \mathbb{E}(e^{-\phi_t})/dt = \left\langle \mathbb{E}_{\omega_{\phi_t}}(\delta/N), d\phi_t/dt \right\rangle, \quad (ii) d\mathcal{E}_{\omega_0}(\phi_t)/dt = \frac{1}{V} \left\langle \omega_{\phi_t}, d\phi_t/dt \right\rangle$$ Moreover, if $\phi_t$ is a $C^0$ -geodesic in $Psh(X,\omega)$ then $$(i')$$ log $\mathbb{E}(e^{-\phi_t})$ is concave, $(ii')$ $\mathcal{E}_{\omega_0}(\phi_t)$ is affine in the real parameter t (note however that $\log \mathbb{E}(e^{-\phi_t})$ is convex along affine curves; compare Remark 2.5 below). The item (i') above follows from the Toeplitz determinant representation 1.6 combined with the positivity results for direct image bundles in [8]. See also the appendix in [5] for another proof of (i') using the structure of determinantal point processes. The key point is the following formula (2.7) $$\partial_t^2 \log \mathbb{E}(e^{-\phi_t}) = \operatorname{Tr}\left(T[\partial_t \partial_{\bar{t}} \phi_t] + \left( (T[\partial_t \phi_t])^2 - T[(\partial_t \phi_t)^2] \right) \right),$$ where Tr denotes the trace and T[f] is the Toeplitz operator with symbol f wrt the perturbed weight $\Phi + \phi_t$ : $$T[f] = \int_X f(y) K_{\Phi + \phi_t}(\cdot, y) (= \Pi_{\Phi + \phi_t}(f \cdot))$$ One then uses the geodesic equation 2.5 to replace $\partial_{\bar{t}}\partial_t\phi_t$ with $|\bar{\partial}_X(\partial_t\phi_t)|^2_{\omega_{\phi_t}}$ in the first term in 2.7 and finally apply the Hörmander-Kodaira $L^2$ -estimate for the inhomogenous $\bar{\partial}_X$ - equation (see 3.16 below) to deduce that $\partial_t^2 \log \mathbb{E}(e^{-\phi_t}) \leq 0$ . 2.1.1. The proof of Proposition 2.2. Now consider the following functional on $C^0(X)$ , which is invariant under addition of constants: $$\mathcal{F}_{\omega}(\phi) := \mathcal{E}_{\omega_0}(\phi) + \frac{1}{N} \log \mathbb{E}(e^{-\phi})$$ For any given $\phi \in C^0(X) \cap \text{Psh}(X,\omega)$ we let $\phi_t$ be the $C^0$ -geodesic such that $\phi_0 = 0$ and $\phi_1 = \phi$ . By the concavity of $\mathcal{F}_{\omega}(\phi_t)$ (resulting from (i') combined with (ii') above) and since $\mathcal{F}_{\omega}(\phi_0) = 0$ we have $$\mathcal{F}_{\omega}(\phi) \le d(\mathcal{F}_{\omega}(\phi_t))/dt_{t=0} = \int (V\mathbb{E}(\delta/N)/\omega - 1)\frac{1}{V}\omega(-d\phi_t/dt)_{t=0}$$ Next, note that, since the inequality in the theorem that we are about to prove is invariant under $\phi \to \phi + C$ we may as well assume that $\sup_X \phi = 0$ . Since $\phi_t$ is convex in t we have $-d\phi_t/dt \le \phi_1 - \phi_0 = \phi$ (we are using right derivatives, which always exist by convexity) and hence $$\mathcal{F}_{\omega}(\phi) \leq \sup_{X} (V\mathbb{E}(\delta/N)/\omega - 1) \frac{1}{V} (\int \omega(-d\phi_t/dt)_{t=0})$$ Next, note that, combining (ii) and (ii') above gives $$\left(\int \omega(-d\phi_t/dt)_{t=0} = d\mathcal{E}_{\omega}(\phi_t)/dt_{t=0} = -\mathcal{E}_{\omega}(\phi)\right)$$ and hence $$\mathcal{F}_{\omega}(\phi) \leq \sup_{X} (V \mathbb{E}(\delta/N)/\omega - 1)(-\mathcal{E}_{\omega}(\phi))$$ Finally, replacing $\phi$ with $\phi - \sup_X \phi$ finishes the proof of the proposition. 2.1.2. The psh projection $P_{\omega}$ . To reduce the case of a general smooth function $\phi$ to an $\omega$ -psh one we will make use of the psh-projection $P_{\omega}$ mapping smooth functions to $\omega$ -psh ones: $$(2.8) (P_{\omega}\phi)(x) := \sup \{ \psi(x) : \psi \in PSH(X,\omega), \psi < \phi \text{ on } X \}$$ It is not hard to see that $P_{\omega}\phi$ is continuous when $\phi$ is and moreover that the following "orthogonality relation" holds [6] (2.9) $$\int_{X} (\phi - P_{\omega}\phi) dd^{c}(P_{\omega}\phi) = 0$$ (as a consequence of the maximum principle for the Laplacian).

Proposition 2.3. Let $(X, \omega)$ be a Riemann surface with a Kähler. Then $$(i) \mathcal{E}_{\omega}(\phi) \le \mathcal{E}_{\omega}(P_{\omega}\phi), \quad (ii) \|d(P_{\omega}\phi)\|_X^2 \le \|d\phi\|_X^2$$ for any $\phi \in C^{\infty}(X)$ .

Proposition 2.4. For any given function $\phi$ on X the following upper bound on the variance of the corresponding linear statistic holds: $$\mathbb{E}(|\tilde{\phi}|^2)/4 \le (1 + \epsilon_N) \|d\phi\|_X^2 + \epsilon_N \|\phi\|_{L^1(X)/\mathbb{R}}^2$$ where $\epsilon_N$ denotes a sequence, independent of $\phi$ , tending to zero. In particular, if $\phi \in L^1(X)$ and $d\phi \in L^2(X)$ then the variance is uniformly bounded from above by a constant independent of N.

Theorem 3.1. Let $L \to X$ be a line bundle over a Riemann surface equipped with a metric $e^{-\Phi}$ with positive curvature form $\omega(=dd^c\Phi)$ such that the Riemannian metric on X defined by $\omega$ has constant scalar curvature R close to x. Then there is a neighbourhood of $\{x\} \times \{x\}$ in $X \times X$ such that the corresponding Bergman kernel $K_k$ satisfies (3.1) $$K_k(z, w) = (k + \frac{1}{2}R)e^{k\psi(\bar{z}, w)} + \epsilon_k$$ where $\psi$ is the local holomorphic function such that $\psi(\bar{z}, w) = \Phi(z)$ and $\epsilon_k$ denotes a smooth section of $kL \boxtimes kL$ whose point-wise norm is of the order $\mathcal{O}(e^{-\delta k})$ . In particular, (3.2) $$B_k(x) := ||K_k(x, x)|| = k + \frac{1}{2}R + \mathcal{O}(e^{-\delta k})$$ (in the case when X is the two-sphere and R is constant on all of X our arguments will give the well-known fact that the error terms vanish identically). Here the Berman kernel $K_k \in H^0(X \times \bar{X}, kL \boxtimes kL)$ denotes the integral kernel of the orthogonal projection from $\mathcal{C}^{\infty}(X, L)$ onto the Hibert space $H^0(X, kL)$ using the $L^2$ -norm defined by the metric on L and volume form $d\nu = \omega$ (formula 1.4). The normalization of R has been chosen so that $R = \deg(TX) (= 2g - 2)$ when it is globally constant (and hence integrating 3.2 against $\omega$ over X gives the Riemann-Roch relation 1.9 for k large). We recall that a sequence $a_k$ is said to be exponentially small, written as $a_k = \mathcal{O}(e^{-k\delta})$ if $|a_k| \leq Ce^{-k\delta}$ for some numbers $C, \delta > 0$ (if $a_k$ are functions then, by definition, the estimate holds uniformly). The case of larger error terms of the form $\mathcal{O}(e^{-(\log k)^2\delta})$ in 3.2 was priouvsly obtained in [31, 32] using Tian's method of peak sections. It was also pointed out there that the case of even larger error terms of the form $\mathcal{O}(k^{-\infty})$ can be deduced from the results in [33] concerning the Yau-Tian-Zelditch-Catlin expansion of $B_k$ , but that one may expect exponentially small error terms (as confirmed in the theorem above). In the case when $L = K_X$ and the scalar curvature is constant on all of X (in particular X then has genus at least two and R < 0) the error term $\mathcal{O}(e^{-\delta k})$ in 3.2 could also be obtained by writing $X = \Gamma/\mathbb{H}$ for a Fuchsian group $\Gamma$ and using that that $B_k$ is constant in the non-compact case setting of $X = \mathbb{H}$ and then estimate the effect of the " $\Gamma$ -periodization" coming from a Poincaré theta series (as pointed out to the author by Steve Zelditch) A similar periodization argument was used in [16] in the case when X is the torus. One motivation to consider the situation when R is not globally constant is to allow applications to the setting of constant curvature metrics with conical singularities and cusps. For example, in the hyperbolic setting this means that the Kähler form $\omega$ is the unique solution to (3.3) $$\operatorname{Ric} \omega = -\omega + \sum_{i} c_{i} \delta_{P_{i}}$$ for given coefficients $c_i \in [0,1] \cap \mathbb{Q}$ and a finite number of points $P_i$ in X ([22], Thm 21.1). Equivalently, $\omega$ has constant scalar curvature -1 on $X - \{P_i\}$ with conical singularities at an angle $2\pi(1-c_i)$ at any $P_i$ such that $c_i < 1$ and a cusp at any $P_i$ such that $c_i = 1$ . Letting $D = \sum_i c_i \mathcal{O}_{P_i}$ be the corresponding $\mathbb{Q}$ — line bundle we then have the following ^&lt;sup>2</sup>The classical case when $X - \{P_i\} = \Gamma/\mathbb{H}$ for a Fuchsian group $\Gamma$ corresponds to the the case when $c_i = 1 - 1/m$ for m a positive integer or infinity and then $\omega$ is induced from the hyperbolic metric on $\mathbb{H}$

Corollary 3.2. Let $L = K_X + D$ and let $\omega$ be the unique (singular) metric on X above. Then the Bergman kernel expansions 3.1 and 3.2 hold for any $x \in X - \{P_i\}$ . Moreover, the positive number $\delta$ appearing in 3.2 may be taken to be arbitrarily close to $$2\log(\cosh(\pi I(x)/\sqrt{2}))$$ where I(x) is the injectivity radius in $X - \{P_i\}$ at x (which coincides with half the length of the shortest closed and simple geodesic on X, passing through x, when D = 0). The Bergman kernel in the previous corollary is, as usual, defined wrt the subspace of $H^0(X, kL)$ consisting of all "cusp forms", i.e. sections vanishing at the cusps and it is well-defined for all k such that $kc_i \in \mathbb{Z}$ for all i. The rest of the section is devoted to the proof of the theorem above; following the scheme in [2] we first prove a local variant of the expansion and then globalize. The main point here is the observation that the local expansion may be obtained using the "local symmetry" of L (as opposed to the general case treated in [2]) which leads to a precise controle of the error terms. As is well-known the local constant curvature condition implies that there exists a local holomorphic coordinate w centered at x on some simply connected neighbourhood U such that $$\omega := \frac{i}{2\pi} 2(1 + R|w|^2)^{-2} dw \wedge d\bar{w} (= dd^c \Phi_0)$$ where $\Phi_0(w) = R^{-1} \log(1 + R|w|^2)$ for $R \neq 0$ and $\Phi_0 = |w|^2$ for R = 0 (obtained in the limit $R \to 0$ ). Now fix a local holomorphic section s of L close to x and write $||s||^2 = e^{-\Phi}$ for a local function $\Phi$ , recalling that $\omega = dd^c\Phi$ (see section 1.5). By the previous relation this means that $\Phi - \Phi_0$ is a harmonic function on U and hence we may write $e^{-\Phi} = |h|^2 e^{-\Phi_0}$ for some non-vanishing holomorphic function h on U. Accordingly, after replacing s with $h^{-1}s$ we may as well assume that we are in the $model\ case\ \Phi = \Phi_0$ . 3.1. Local Bergman kernels for the model cases. Given a smooth function $\Phi$ on a domain U in $\mathbb C$ containing 0 we let $$\langle f, g \rangle_{U, k\Phi} := \int_U f \bar{g} e^{-k\Phi} dd^c \Phi$$ and denote by $H_{k\Phi}(U)$ the space of all holomorphic functions on U such $||f||_{U,k\Phi}^2$ (:= $\langle f,g\rangle_{U,k\Phi}\rangle < \infty$ . Following [2] we will say that $K_{(k)}(z,\zeta)$ is a (local) Bergman kernel mod $\mathcal{O}(e^{-k\delta})$ (with respect to $\Phi$ ) if it is holomorphic in $\zeta$ and there exists number $\delta > 0$ such that for any $f \in H_{k\Phi}(U)$ we have, for all z in some neighbourhood $V \subset U$ of 0 that (3.4) $$f_k(z) = \left\langle f_k, \chi K_{(k)}(z, \cdot) \right\rangle_{U, k\Phi} + \|f\|_{U, k\Phi} \mathcal{O}(e^{-k\delta}) e^{k\Phi/2}$$ where $\chi$ denotes a smooth function $\chi$ compactly supported on U which is equal to one on $\frac{1}{2}U$ .

Proposition 3.3. Let $\Phi(w) = -2\log(1+R|w|^2)/R$ . Then the function $K_{(k)}(z,\zeta) = (k+\frac{R}{2})(1+R\zeta\bar{z})^{2k/R}$ is a local Bergman kernel mod $\mathcal{O}(e^{-k\delta})$ (wrt $\Phi$ ) when $R \neq 0$ and $K_{(k)}(z,\zeta) = ke^{\bar{z}\zeta}$ when R = 0 (coinciding with the limit when $R \to 0$ ).

Proposition 3.4. Let $L \to X$ be a positive Hermitian holomorphic line bundle over a compact complex manifold X and let $x \in X$ be a fixed point such that the local weight $\Phi$ of the metric wrt some trivialization of L around x is real-analytic ad admits a local Bergman kernel $K_{(k)}$ mod $\mathcal{O}(e^{-k\delta})$ such that (3.13) $$K_{(k)}(z,\zeta) = a_k e^{k\psi(z,\bar{\zeta})}$$ for some sequence $a_k$ with sub-exponential growth (i.e. $|a_k| \leq C_{\delta}e^{k\delta}$ for any $\delta > 0$ ). Then the (global) Bergman kernel $K_k$ associated to kL satisfies the uniform estimate $$\left\| K_k - K_{(k)} \right\|_{k\Phi} \le C e^{-\delta k}$$ on some neighbourhood $U \times U$ of $\{x\} \times \{x\}$ for some numbers $C, \delta > 0$ .