\chapter{The coarse eigenvector law}\label{s:coarse}

In this section, we prove \zcref{t:coarse,c:coarse-cov,t:shape,c:mean-shape}, and, in \zcref{p:coarse-endpoints}, the rows of \zcref{p:endpoints} that concern the mass measure and the eigenvector correlation. The argument has four steps. First, we compute all fixed moments $N^k \langle \prod_a w_\alpha(i_a) \rangle_{E, r}$ at separated interior sites (\zcref{t:coarse-moments}). For this we use a positive source built from permanents of $\Im G$, whose limit at fixed damping is the permanent pairing of \zcref{d:res-chain}, by \zcref{t:interface}(b); its residue at zero damping is given by \zcref{l:res-permanent}, and \zcref{t:extract-main} converts it into the moment. This is the main step, and the remaining ones are soft. Second, Scheff\'e's lemma upgrades the pointwise limits to $L^1$ limits of the step densities of the moments, including the collision and boundary strips that the first step excludes. Third, the Stone--Weierstrass theorem identifies the law of the mass measure. Finally, the rooted laws, the infinite-wire shape, and its mean follow from the Cameron--Martin theorem, Brownian scaling, Dufresne's identity, and the Kontorovich--Lebedev transform. In addition, \zcref{l:coarse-second-moment} computes the second moment of the mean shape, which is used only in \zcref{r:coarse-mean-physics}, where we compare our results with the physics literature.

The coarse law was derived in the physics literature for a finite disordered wire in the unitary class. Mirlin \cite[(3.32)--(3.34)]{mirlin2000statistics}, building on Fyodorov and Mirlin \cite{fyodorov1994statistical} and on Kolokolov's method for strictly one-dimensional systems \cite{kolokolov1993method}, wrote the multipoint eigenfunction statistics as a Liouville path integral and gave the law of the envelope of an eigenfunction. In this description, an eigenfunction is the product of a fast factor with Gaussian statistics and a smooth envelope \cite[(3.33)]{mirlin2000statistics}. Positions along the wire are measured in units of the localization length, so that the wire is the interval $[0, X]$. Following \cite[(3.34)]{mirlin2000statistics}, we let $\theta(t)$ denote the logarithm of the square of the envelope at the position $t \in [0, X]$. In particular, $\theta(0)$ and $\theta(X)$ are its values at the two ends of the wire. The law of the path $\theta$ in \cite[(3.34)]{mirlin2000statistics} has, formally, the density
\begin{equation}\label{e:coarse-mirlin}
 e^{-X/4} e^{-(\theta(0) + \theta(X))/2} \exp\Big( -\frac14 \int_0^X \dot\theta(t)^2 \dd t \Big)
\end{equation}
with respect to a flat measure on paths, restricted by the normalization $X^{-1} \int_0^X e^{\theta(t)} \dd t = 1$ of \cite[(3.32)]{mirlin2000statistics}. The Gaussian factor in \eqref{e:coarse-mirlin} describes $\sqrt 2$ times a Brownian motion, which is the diffusion with generator $\partial_\theta^2$, started from a point $\theta(0) \in \R$ that carries Lebesgue measure.

We translate this law into our variables in two steps. First, we factor the weight in front of the Gaussian factor in \eqref{e:coarse-mirlin} as
\[
 e^{-X/4} e^{-(\theta(0) + \theta(X))/2} = e^{-\theta(0)} \cdot e^{-X/4} e^{-(\theta(X) - \theta(0))/2} .
\]
The second factor is the Girsanov density, on the time interval $[0, X]$, of the drift $-1$ for the diffusion with generator $\partial_\theta^2$. After this change of measure, $(\theta(t) - \theta(0))_{t \in [0, X]}$ has the law of $(\sqrt 2 \beta_t - t)_{t \in [0, X]}$ for a standard Brownian motion $\beta$. By Brownian scaling, $(\theta(Xs) - \theta(0))_{s \in [0, 1]}$ then has the law of the process $g_X$ of \zcref{d:brownian}. Second, since $X^{-1} \int_0^X e^{\theta(t)} \dd t = e^{\theta(0)} \int_0^1 e^{\theta(Xs) - \theta(0)} \dd s$, the normalization constraint reads $e^{\theta(0)} Z_X = 1$, and the integration of $e^{-\theta(0)}$ over $\theta(0)$ against this constraint, after the substitution $u = e^{\theta(0)}$, produces the weight $Z_X$. On this constraint, the squared envelope at the position $Xs$ satisfies $e^{\theta(Xs)} = e^{g_X(s)}/Z_X$, which is the density of $\mu_X$, and the envelope law becomes the law of $\mu_X$ under $\mathbb Q_X$. The rooted shape at large $X$ is implicit in the same formulas \cite[(3.30)--(3.34)]{mirlin2000statistics}, and the tail of the mean shape, with the constant $\pi^{7/2}/32$, is due to Efetov and Larkin \cite[(56)]{efetov1983kinetics}. Our results give proofs of these predictions for the Gaussian band matrix.

The same law is known rigorously for the critical one-dimensional random Schr\"odinger operators recalled in \zcref{s:intro-background}, and Rifkind and Vir\'ag conjectured that their eigenvector shape is universal for critical one-dimensional random operators \cite[Conjectures~1.2 and~1.3]{rifkind2018eigenvectors}. We show in \zcref{r:coarse-rv} that, with their parameter $\tau$ equal to $4X$, their limit \cite[Theorem~1.1]{rifkind2018eigenvectors} is the law of $\mu_X$ under $\mathbb Q_X$, and at the start of \zcref{s:coarse-wire} that $p_\infty$ is the crossover shape of Dumaz and Labb\'e \cite[Theorem~3]{dumaz2023localization}, up to a linear change of the space variable and its centering; \zcref{t:coarse,t:shape} therefore add Gaussian band matrices at the critical bandwidth to the operators that this shape describes.

\paragraph{Conventions.} We keep the conventions of \zcref{s:extract}. In particular, $E_* < E_{\mathrm{mid}} < E_{\mathrm{out}}$ are the energies of \eqref{e:iface-energies}, and statements hold along sequences $N, W \to \infty$ with $N/W^2$ in a compact subset of $(0, \infty)$. A damping $\eps > 0$ corresponds to $z = 2\pi\eps$, and we set $z_+ = E + \iu \eps/a_N$, which is the energy $z^{\mathrm B}_+$ of \eqref{e:iface-Z}. For a matrix $A$, we let $\Im A = (A - A^*)/(2\iu)$; in particular, $\Im G(z_+)$ is positive definite. The cell of a site $j \in \{1, \dots, N\}$ is the interval $I_{j, N} = ((j - 1)/N, j/N]$.

\section{Permanent moment measures}\label{s:coarse-permanent}

The products $\prod_a w_\alpha(i_a)$ are the diagonal part of a positive moment measure, which we now define. Recall that the permanent of a $k \times k$ matrix $A$ is
\[
 \operatorname{perm} A = \sum_{\pi \in S_k} \prod_{a = 1}^k A_{a \pi(a)} .
\]

\begin{definition}\label{d:coarse-permanent}
Let $k \ge 1$, and let $i_1, \dots, i_k$ be distinct sites. We define the \emph{permanent moment measure} at these sites to be the random measure on $\R^k$
\[
 \bm\Pi_N = \frac{N^k}{k!} \sum_{\alpha_1, \dots, \alpha_k = 1}^N \big| \operatorname{perm} [u_{\alpha_b}(i_a)]_{a, b = 1}^k \big|^2 \, \delta_{(x_{\alpha_1}, \dots, x_{\alpha_k})} .
\]
\end{definition}

\begin{lemma}\label{l:coarse-permanent}
Let $k \ge 1$ and let $i_1 < \dots < i_k$ be sites.
\begin{enumerate}[label=(\alph*)]
\item The measure $\bm\Pi_N$ is an eigenvector-weighted moment measure of order $k$ in the sense of \zcref{d:extract-measures}, and it is dominated at $i_1, \dots, i_k$ with constant $C_0 = 1$. With $r_k = k!$, its diagonal part is given by
\[
 \bm\Pi_N^{\mathrm d} = N^k \sum_{\alpha = 1}^N \prod_{a = 1}^k w_\alpha(i_a) \, \delta_{x_\alpha} .
\]
\item For all $\eps > 0$, with $\Pi_N = \E \bm\Pi_N$,
\begin{equation}\label{e:coarse-permanent-transform}
 \int_{\R^k} \prod_{b = 1}^k P_\eps(t_b) \, \Pi_N(\dd \mathbf t) = s_E^{-k} \, \E \operatorname{perm} \big[ (\Im G(z_+))_{i_a i_b} \big]_{a, b = 1}^k .
\end{equation}
\item Let $\varrho = (1, 1)^{\mathrm T}$ and $v^{\mathrm T} = \frac\iu2 (1, -1)$, and let $Y$ be the local source of \zcref{d:iface-local} with source rank one, that is, the even supermatrix with $2 \times 2$ blocks, with $Y^{\mathrm F} = 0$ and boson--boson block
\[
 Y^{\mathrm B} = -\varrho v^{\mathrm T} = -\frac\iu2 \begin{pmatrix} 1 & -1 \\ 1 & -1 \end{pmatrix}
\]
in the order $(\mathrm R, \mathrm A)$ of the flavors. (The block $Y^{\mathrm B}$ is also a matrix of rank one.) Then, with the local source function $T_N^{(1)}$ of \zcref{d:iface-local}, of source rank one, at $\mathbf u = 0$ and $\mathbf b = \eps$,
\begin{equation}\label{e:coarse-permanent-source}
 \E \operatorname{perm} \big[ (\Im G(z_+))_{i_a i_b} \big]_{a, b = 1}^k
 = \partial_{t_1} \cdots \partial_{t_k} T_N^{(1)}(0, \eps; t_1 Y, \dots, t_k Y) \big|_{t = 0} .
\end{equation}
\end{enumerate}
\end{lemma}

\begin{proof}
We begin with (a). Let
\[
 c_N(\alpha) = \frac{N^k}{k!} \big| \operatorname{perm}[u_{\alpha_b}(i_a)] \big|^2
\]
denote the coefficient of the point mass at $(x_{\alpha_1}, \dots, x_{\alpha_k})$. Multiplying $u_\alpha$ by a phase multiplies a column of the matrix by this phase and leaves $c_N(\alpha)$ unchanged. Thus $c_N(\alpha)$ is a function of the eigenprojections. If all labels are equal to $\alpha$, then all columns are equal, and
\[
 \operatorname{perm}[u_\alpha(i_a)]_{a, b = 1}^k = k! \prod_{a = 1}^k u_\alpha(i_a) , \qquad c_N(\alpha, \dots, \alpha) = k! \, N^k \prod_{a = 1}^k w_\alpha(i_a) .
\]
This is the diagonal part stated in (a).

For the domination, we apply the Cauchy--Schwarz inequality to the $k!$ terms of the permanent, and we find
\[
 \frac{1}{k!} \big| \operatorname{perm}[u_{\alpha_b}(i_a)] \big|^2 \le \sum_{\pi \in S_k} \prod_{a = 1}^k w_{\alpha_{\pi(a)}}(i_a) .
\]
Let $f_1, \dots, f_k \ge 0$. We multiply by $N^k \prod_b f_b(x_{\alpha_b})$ and sum over $\alpha_1, \dots, \alpha_k$. For fixed $\pi$, the sum factorizes into $\prod_a \ell_{i_a, N}(f_{\pi(a)})$, by the definition \eqref{e:extract-local-measure} of $\ell_{i, N}$. This is \eqref{e:extract-domination} with $C_0 = 1$.

For (b), let $c_\alpha \ge 0$, and consider the matrix
\[
 K = \sum_\alpha c_\alpha u_\alpha u_\alpha^* .
\]
Expanding the permanent of $[K_{i_a i_b}]$, we have
\[
 \operatorname{perm} [K_{i_a i_b}] = \sum_{\alpha_1, \dots, \alpha_k} \prod_a c_{\alpha_a} u_{\alpha_a}(i_a) \sum_{\pi \in S_k} \prod_a \overline{u_{\alpha_a}(i_{\pi(a)})} .
\]
The inner sum is the complex conjugate of $\operatorname{perm}[u_{\alpha_b}(i_a)]$, which is invariant under permutations of the labels $\alpha_1, \dots, \alpha_k$. Averaging the factor $\prod_a u_{\alpha_a}(i_a)$ over these permutations turns it into $\operatorname{perm}[u_{\alpha_b}(i_a)]/k!$. We obtain the permanental Cauchy--Binet identity
\[
 \operatorname{perm} [K_{i_a i_b}] = \frac{1}{k!} \sum_{\alpha_1, \dots, \alpha_k} \prod_b c_{\alpha_b} \, \big| \operatorname{perm}[u_{\alpha_b}(i_a)] \big|^2 .
\]
The spectral theorem and \eqref{e:scaling} give
\[
 \Im G(z_+) = \sum_\alpha \frac{\eps/a_N}{(\lambda_\alpha - E)^2 + (\eps/a_N)^2} \, u_\alpha u_\alpha^* = s_E N \sum_\alpha P_\eps(x_\alpha) \, u_\alpha u_\alpha^* ,
\]
where we used $\pi a_N = s_E N$. Applying the identity with $c_\alpha = s_E N P_\eps(x_\alpha)$ and taking expectations, we obtain \eqref{e:coarse-permanent-transform}.

For (c), we first recall the matrices $Z_{\mathrm B}$ and $Z_{\mathrm F}$ of \eqref{e:iface-Z}, which for source rank one are the diagonal $2 \times 2$ matrices
\[
 Z_{\mathrm B} = \diag(z^{\mathrm B}_+, z^{\mathrm B}_-) , \qquad Z_{\mathrm F} = \diag(z^{\mathrm F}_{1, +}, z^{\mathrm F}_{1, -})
\]
in the flavors $(\mathrm R, \mathrm A)$, with $z^{\mathrm B}_\pm = E \pm \iu \eps/a_N$ and $z^{\mathrm F}_{1, \pm} = E + (u_1 \pm \iu b_1)/a_N$. In particular, at $(\mathbf u, \mathbf b) = (0, \eps)$, we have
\[
 Z_{\mathrm B} = Z_{\mathrm F} = \diag(z_+, \bar z_+) .
\]

We set
\[
 A = \sum_{a = 1}^k t_a e_{i_a} e_{i_a}^* , \qquad M = \Id_N \otimes Z_{\mathrm B} - H \otimes \Id_2 ,
\]
where, as in \eqref{e:iface-local}, the first factor of each tensor product acts on the sites and the second on the flavors. With the matrix units $E_{\mathrm{RR}}$ and $E_{\mathrm{AA}}$ of the flavors $\mathrm R$ and $\mathrm A$, we have
\[
 M = (z_+ - H) \otimes E_{\mathrm{RR}} + (\bar z_+ - H) \otimes E_{\mathrm{AA}} , \qquad
 M^{-1} = -G(z_+) \otimes E_{\mathrm{RR}} - G(\bar z_+) \otimes E_{\mathrm{AA}} .
\]
The local source $t_a Y$ at the site $i_a$ has the BB block $-t_a \varrho v^{\mathrm T}$, and therefore
\[
 \sum_{a = 1}^k e_{i_a} e_{i_a}^* \otimes t_a Y^{\mathrm B} = - A \otimes \varrho v^{\mathrm T} = -(A \otimes \varrho)(\Id_N \otimes v^{\mathrm T}) ,
\]
where $A \otimes \varrho$ is a $2N \times N$ matrix and $\Id_N \otimes v^{\mathrm T}$ is an $N \times 2N$ matrix.

Since $Y^{\mathrm F} = 0$ and $Z_{\mathrm F} = Z_{\mathrm B}$, the ratio in \eqref{e:iface-local} reduces to
\[
 \frac{\det M}{\det(M - A \otimes \varrho v^{\mathrm T})} .
\]

We now apply Sylvester's determinant identity $\det(\Id_{2N} - V V') = \det(\Id_N - V' V)$ with $V = M^{-1}(A \otimes \varrho)$ and $V' = \Id_N \otimes v^{\mathrm T}$. Writing $\varrho = (\varrho_{\mathrm R}, \varrho_{\mathrm A})^{\mathrm T} = (1, 1)^{\mathrm T}$ and $v = (v_{\mathrm R}, v_{\mathrm A})^{\mathrm T} = (\iu/2, -\iu/2)^{\mathrm T}$, we find
\begin{align*}
 \frac{\det(M - A \otimes \varrho v^{\mathrm T})}{\det M} &= \det\big( \Id_N - (\Id_N \otimes v^{\mathrm T}) M^{-1} (A \otimes \varrho) \big) \\
 &= \det\big( \Id_N + v_{\mathrm R} \varrho_{\mathrm R} \, G(z_+) A + v_{\mathrm A} \varrho_{\mathrm A} \, G(\bar z_+) A \big) \\
 &= \det\big( \Id_N + \tfrac\iu2 \big( G(z_+) - G(\bar z_+) \big) A \big) = \det\big( \Id_N - \Im G(z_+) \, A \big) ,
\end{align*}
where in the second step we used $(\Id_N \otimes v^{\mathrm T})(G \otimes E_{\mathrm{RR}})(A \otimes \varrho) = v_{\mathrm R} \varrho_{\mathrm R} \, G A$ and its analogue for $E_{\mathrm{AA}}$, and in the last step $G(\bar z_+) = G(z_+)^*$. Since $\det(\Id_N - \Im G(z_+) A) = \det(\Id_N - A \Im G(z_+))$ by Sylvester's identity, the ratio in \eqref{e:iface-local} reduces to $\det(\Id_N - A \Im G(z_+))^{-1}$.

Conditionally on $H$, let $\zeta = (\zeta_1, \dots, \zeta_N)$ be a centered complex Gaussian vector with
\[
 \E[\zeta \zeta^* \mid H] = \Im G(z_+) , \qquad \E[\zeta \zeta^{\mathrm T} \mid H] = 0 .
\]
For small $t$, the Gaussian integral gives
\[
 \det\big( \Id_N - A \Im G(z_+) \big)^{-1} = \E \Big[ \exp\Big( \sum_a t_a |\zeta_{i_a}|^2 \Big) \Bigm| H \Big] .
\]
We differentiate at $t = 0$ and use Wick's theorem for complex Gaussian vectors. We find that $\partial_{t_1} \cdots \partial_{t_k}$ of the ratio at $t = 0$ is
\[
 \E \Big[ \prod_a |\zeta_{i_a}|^2 \Bigm| H \Big] = \operatorname{perm}\big[ (\Im G(z_+))_{i_a i_b} \big] .
\]
As noted after \zcref{d:iface-local}, the derivatives may be taken under the expectation, and \eqref{e:coarse-permanent-source} follows.
\end{proof}

We now identify the limit of \eqref{e:coarse-permanent-transform}, using \zcref{t:interface}(b). This is the only step of this section that works with the sigma model directly; the other steps use only its residues, computed in \zcref{s:residue}. We first recall the objects involved. The field $Q$ of \zcref{d:sigma-target}, at rank one, is a $(2 | 2)$ supermatrix of superfunctions on $\mathcal M_1$ with $Q^2 = \Id$, and $Q^{\mathrm{BB}}_{\mathrm{RA}}$ denotes the entry of its boson--boson block in the retarded row and the advanced column. Recall the supermatrix $\Lambda = \diag(\Lambda_{\mathrm B} \mid \Lambda_{\mathrm F})$ of \zcref{d:rep-sources}, which at rank one has the blocks $\Lambda_{\mathrm B} = \Lambda_{\mathrm F} = \diag(1, -1)$ in the flavors $(\mathrm R, \mathrm A)$. The superfunction $d_1(Q) = \frac12 \Str(\Lambda Q)$ is the damping term of the generator $\mathsf G_{X, z} = X \mathsf C_1 + z \, d_1$ of \eqref{e:sigma-generator}, and in the polar coordinates of \zcref{s:sigma-polar}, we have $d_1 = B - F$, by \zcref{l:sigma-polar}\ref{i:sigma-polar-entries}. Let $e^{-t \mathsf G_{X, z}}$ be the semigroup of \zcref{p:sigma-regular}, and let $\mathcal B_1$ be the Berezin pairing of \zcref{d:sigma-integral}. For $0 \le x_1 \le \dots \le x_k \le 1$, the permanent pairing of \zcref{d:res-chain} is
\begin{equation}\label{e:coarse-pairing-recall}
 \mathfrak p^{X, z}_k(x_1, \dots, x_k) = \mathcal B_1 \big( e^{-x_1 \mathsf G_{X, z}} 1, \, \mathfrak s \, e^{-(x_2 - x_1) \mathsf G_{X, z}} \mathfrak s \cdots e^{-(x_k - x_{k - 1}) \mathsf G_{X, z}} \mathfrak s \, e^{-(1 - x_k) \mathsf G_{X, z}} 1 \big) ,
\end{equation}
where $\mathfrak s$ is the source \eqref{e:res-source},
\[
 \mathfrak s(Q) = \tfrac12 \big( Q^{\mathrm{BB}}_{\mathrm{RR}} + Q^{\mathrm{BB}}_{\mathrm{RA}} - Q^{\mathrm{BB}}_{\mathrm{AR}} - Q^{\mathrm{BB}}_{\mathrm{AA}} \big) .
\]
Finally, recall the orbit resolvent $\mathcal G(Q) = -\kappa_E \Id + \iu s_E Q$ of \eqref{e:iface-orbit-resolvent}, and the orbit multiplier $\mathfrak j_Y(Q) = \Sdet(\Id - \mathcal G(Q) Y)^{-1}$ of a local source $Y$, whose Taylor coefficients $\mathfrak j^{(\alpha)}$ in \eqref{e:iface-multiplier} are the multipliers of \zcref{t:interface}(b). Here $\Sdet$ is the superdeterminant of \zcref{d:grass-supermatrix}. For an even supermatrix $M$ with blocks $A, B, C, D$, this definition gives, for small $t$,
\begin{equation}\label{e:coarse-sdet-derivative}
\begin{gathered}
 \Sdet(\Id + tM) = \frac{\det\big( \Id + tA - t^2 B (\Id + tD)^{-1} C \big)}{\det(\Id + tD)} , \\
 \partial_t \Sdet(\Id + tM)^{-1} \big|_{t = 0} = -\Tr A + \Tr D = -\Str M .
\end{gathered}
\end{equation}

\begin{proposition}\label{p:coarse-source-limit}
Let $k \ge 1$ and $\eps > 0$, and let $|E| \le E_{\mathrm{out}}$, with $E_{\mathrm{out}}$ as in \eqref{e:iface-energies}. Let $i_1 < \dots < i_k$ be sites with $\delta \le i_a/N \le 1 - \delta$ and $i_a/N \to x_a$, and suppose that $X_N(E) \to X \in (0, \infty)$. Then
\[
 \int_{\R^k} \prod_{b = 1}^k P_\eps(t_b) \, \Pi_N(\dd \mathbf t) \longrightarrow \mathfrak p^{X, 2\pi\eps}_k(x_1, \dots, x_k) .
\]
\end{proposition}

\begin{proof}
Passing to subsequences, we may assume that $E$ converges. Let $Y$ be the local source of \zcref{l:coarse-permanent}(c). We begin with its multiplier.

Since $Y^{\mathrm B} = -\varrho v^{\mathrm T}$ and $Y^{\mathrm F} = 0$, we have $\Id - t \mathcal G(Q) Y = \Id + t M$ with $M = \mathcal G(Q) \diag(\varrho v^{\mathrm T} \mid 0)$. The supermatrix $M$ has the BB block $\mathcal G(Q)^{\mathrm{BB}} \varrho v^{\mathrm T}$ and a vanishing FF block. Using \eqref{e:coarse-sdet-derivative}, we find
\begin{equation}\label{e:coarse-multiplier}
 \partial_t \mathfrak j_{tY}(Q) \big|_{t = 0} = -\Tr\big( \mathcal G(Q)^{\mathrm{BB}} \varrho v^{\mathrm T} \big) = -v^{\mathrm T} \big( -\kappa_E \Id + \iu s_E Q^{\mathrm{BB}} \big) \varrho = s_E \, \mathfrak s(Q) ,
\end{equation}
where in the last step we used $v^{\mathrm T} \varrho = 0$ and
\[
 v^{\mathrm T} Q^{\mathrm{BB}} \varrho = \tfrac\iu2 \big( Q^{\mathrm{BB}}_{\mathrm{RR}} + Q^{\mathrm{BB}}_{\mathrm{RA}} - Q^{\mathrm{BB}}_{\mathrm{AR}} - Q^{\mathrm{BB}}_{\mathrm{AA}} \big) = \iu \, \mathfrak s(Q) .
\]

We next apply \zcref{t:interface}(b) at source rank one, with the $m = k$ sites $i_1 < \dots < i_k$ and at the base point $(\mathbf u, \mathbf b) = (0, \eps)$. By the chain rule, $\partial_{t_a}$ at $t = 0$ is the derivative $\sum_e Y_e \, \partial_{(Y_a)_e}$ in the direction $Y$ of the local source at $i_a$, where $e$ runs over the entries of the BB block. We expand $\partial_{t_1} \cdots \partial_{t_k}$ in this way, and we apply \eqref{e:iface-jets} to each term, with first-order multi-indices. The limits are pairings that are linear in each multiplier, and by \eqref{e:coarse-multiplier}, their sum has the multipliers
\[
 \sum_e Y_e \, \mathfrak j^{(e)} = \partial_t \mathfrak j_{tY} \big|_{t = 0} = s_E \mathfrak s .
\] The base point corresponds to the source supermatrix $\Gamma = \eps \Lambda$ of \zcref{d:rep-sources}, with $\Lambda = \diag(1, -1 \mid 1, -1)$ as recalled above. By \eqref{e:sigma-potential}, the generator of \zcref{t:interface}(b) satisfies
\[
 \mathsf G_{X, \Gamma} = X \mathsf C_1 + 2\pi\eps \, d_1 = \mathsf G_{X, 2\pi\eps} .
\] Writing $\mathsf G = \mathsf G_{X, 2\pi\eps}$, we conclude that \zcref{t:interface}(b) yields
\begin{equation}\label{e:coarse-iface-limit}
 \partial_{t_1} \cdots \partial_{t_k} T_N^{(1)}(0, \eps; t_1 Y, \dots, t_k Y) \big|_{t = 0}
 \longrightarrow s_E^k \, \mathcal B_1 \big( 1, \, e^{-x_1 \mathsf G} \mathfrak s \, e^{-(x_2 - x_1) \mathsf G} \mathfrak s \cdots \mathfrak s \, e^{-(1 - x_k) \mathsf G} 1 \big) .
\end{equation}

We now show that the pairing on the right is the permanent pairing. Let
\[
 f = \mathfrak s \, e^{-(x_2 - x_1) \mathsf G} \mathfrak s \cdots \mathfrak s \, e^{-(1 - x_k) \mathsf G} 1 .
\]
Since $x_k \le 1 - \delta$, the superfunction $e^{-(1 - x_k) \mathsf G} 1$ is rapidly decreasing, by \zcref{p:sigma-regular}\ref{i:sigma-regular-weights}. The source $\mathfrak s$ is a multiplier of degree one, by \zcref{l:sigma-multipliers}\ref{i:sigma-mult-Q}, and multiplication by $\mathfrak s$ maps $L^2_p$ to $L^2_{p - 1}$ for all $p$, by \zcref{l:grass-norm}\ref{i:grass-norm-sub} and the definition \eqref{e:sigma-Lp} of the norm of $L^2_p$. The semigroup maps each $L^2_p$ to itself, and hence $f \in L^2_p$ for all $p$. By \zcref{p:sigma-regular}\ref{i:sigma-regular-split}, we also have $1 \in L^2_{-q}$ for $q > q_1/2$, where $q_1$ is the exponent of the volume bound in \zcref{l:sigma-body}\ref{i:sigma-body-volume}. The transpose symmetry \zcref{p:sigma-regular}\ref{i:sigma-regular-transpose}, applied with $p = -q$, the superfunction $1 \in L^2_{-q}$, and $f \in L^2_q$, gives
\[
 \mathcal B_1 \big( 1, e^{-x_1 \mathsf G} f \big) = \mathcal B_1 \big( e^{-x_1 \mathsf G} 1, f \big) = \mathfrak p^{X, 2\pi\eps}_k(x_1, \dots, x_k) ,
\]
where the last equality is \eqref{e:coarse-pairing-recall} with $z = 2\pi\eps$. Finally, combining this identity with \eqref{e:coarse-iface-limit}, \eqref{e:coarse-permanent-transform}, and \eqref{e:coarse-permanent-source}, we have
\begin{align*}
 \int_{\R^k} \prod_{b = 1}^k P_\eps(t_b) \, \Pi_N(\dd \mathbf t)
 &= s_E^{-k} \, \partial_{t_1} \cdots \partial_{t_k} T_N^{(1)}(0, \eps; t_1 Y, \dots, t_k Y) \big|_{t = 0} \\
 &\longrightarrow s_E^{-k} \cdot s_E^k \, \mathfrak p^{X, 2\pi\eps}_k(x_1, \dots, x_k) = \mathfrak p^{X, 2\pi\eps}_k(x_1, \dots, x_k) . \qedhere
\end{align*}
\end{proof}

\section{All fixed moments}\label{s:coarse-moments}

We first record the properties of the limiting moment densities that we use.

\begin{lemma}\label{l:coarse-densities}
Let $k \ge 1$. The function $(X, \mathbf x) \mapsto M_k^X(\mathbf x)$ is positive and continuous on $(0, \infty) \times [0, 1]^k$, and symmetric in $\mathbf x$. Moreover, $M_1^X = 1$, and for $k \ge 2$,
\begin{equation}\label{e:coarse-consistency}
 \int_0^1 M_k^X(x_1, \dots, x_{k - 1}, y) \dd y = M_{k - 1}^X(x_1, \dots, x_{k - 1}) , \qquad \int_{[0, 1]^k} M_k^X(\mathbf x) \dd \mathbf x = 1 .
\end{equation}
\end{lemma}

\begin{proof}
All the assertions follow from the definition of $M_k^X$ in \zcref{d:brownian}, by dominated convergence and Tonelli's theorem. We realize all $g_X$ with one Brownian motion $\beta$ (with $X$ as a parameter in \zcref{d:brownian}), and the map $(X, t) \mapsto g_X(t)$ is continuous. For $X \le X_+$, as in the proof of \zcref{l:res-hierarchy}, the integrand in the definition of $M_k^X$ is bounded by
\[
 \exp\Big( (2k - 1) \big( \sqrt{2X_+} \sup_{t \le 1} |\beta_t| + X_+ \big) \Big) ,
\]
which is integrable. The continuity follows by dominated convergence. Positivity and symmetry are clear, and $\E e^{g_X(x)} = 1$ implies $M_1^X = 1$. Integrating $e^{g_X(y)}$ over $y \in [0, 1]$ gives $Z_X$, which proves \eqref{e:coarse-consistency}.
\end{proof}

\begin{theorem}\label{t:coarse-moments}
Let $k \ge 1$, $\delta \in (0, 1/2)$, and let $\bar r_N \to 0$ be a deterministic sequence. Then, uniformly in the critical regime, in $0 < r \le \bar r_N$, and in sites $i_1, \dots, i_k$ with $\delta \le i_a/N \le 1 - \delta$ and $|i_a - i_b| \ge \delta N$ for $a \ne b$,
\begin{equation}\label{e:coarse-moments}
 N^k \Big\langle \prod_{a = 1}^k w_\alpha(i_a) \Big\rangle_{E, r} - M_k^{X_N(E)}(i_1/N, \dots, i_k/N) \longrightarrow 0 .
\end{equation}
\end{theorem}

\begin{proof}
We apply \zcref{c:extract-windows} with
\[
 \chi = \phi , \qquad F_\alpha = \prod_a w_\alpha(i_a) , \qquad \bar c_N(e) = M_k^{X_N(e)}(i_1/N, \dots, i_k/N) .
\]
The parameter $X_N(e) = N/((4 - e^2) W^2)$ ranges over a compact subset of $(0, \infty)$ and is Lipschitz in $e$, uniformly. Together with \zcref{l:coarse-densities}, this shows that the functions $\bar c_N$ are bounded and equicontinuous on $[-E_{\mathrm{mid}}, E_{\mathrm{mid}}]$, uniformly in $N$ and in the sites.

It remains to verify the hypothesis of \zcref{c:extract-windows}. Recall that $\chi = \phi$ is the window of \zcref{d:selection}, and that $f_{N, e}$ is the density of the measure \eqref{e:extract-unfolded-at-e}, which here is
\[
 N^k \E \sum_\alpha \prod_{a = 1}^k w_\alpha(i_a) \, \delta_{a_N(e)(\lambda_\alpha - e)} ,
\]
the expected diagonal part of the permanent moment measure of \zcref{d:coarse-permanent} at the sites $i_1, \dots, i_k$, unfolded at the energy $e$ (\zcref{l:coarse-permanent}(a)). The hypothesis of \zcref{c:extract-windows} is
\[
 \sup_{|e| \le E_{\mathrm{mid}}} |f_{N, e}(0) - \bar c_N(e)| \to 0 ,
\]
uniformly in the sites.

We derive this hypothesis from \zcref{r:extract-uniform}, with the energy and the sites as parameters. Thus we take a sequence of energies $|e_N| \le E_{\mathrm{mid}}$ and of admissible sites, and we show that, along a subsequence, the densities $f_{N, e_N}$ converge locally uniformly to a constant $c_*$, and that $\bar c_N(e_N) \to c_*$. Both statements are to hold uniformly in the sites.

For $k = 1$, the measure in question is the local spectral measure $\ell_{i_1, N}$, unfolded at $e_N$. Here \zcref{c:extract-onepoint} applies with $c_* = 1 = M_1^X$, since $\E \ell^\eps_{i_1, N}(0) \to 1$ at interior sites for all $|e_N| \le E_{\mathrm{out}}$, by \zcref{p:int-balanced}.

Let $k \ge 2$. We pass to a subsequence along which $X_N(e_N)$ converges to some $X \in (0, \infty)$ and $i_a/N \to x_a$, and we order the sites. Then $\delta \le x_a \le 1 - \delta$ and $|x_a - x_b| \ge \delta$. We apply \zcref{t:extract-main} at the energy $e_N$ to the permanent moment measure of \zcref{d:coarse-permanent}, with $r_k = k!$.

The hypotheses of \zcref{t:extract-main} hold for the following reasons. \zcref[S]{l:coarse-permanent}(a) shows that the measure is dominated at the sites with $C_0 = 1$ and has the diagonal part above. The source limit \eqref{e:extract-source} holds with $\mathfrak f(z) = \mathfrak p^{X, z}_k(x_1, \dots, x_k)$, by \zcref{p:coarse-source-limit}. Finally, \zcref{l:res-permanent} and \eqref{e:extract-constant} give
\[
 \lim_{z \downarrow 0} z^{k - 1} \mathfrak f(z) = k (2k - 3)!! \, M_k^X(x_1, \dots, x_k) = \varkappa_{k, k!} \, M_k^X(x_1, \dots, x_k) .
\]

Thus \zcref{t:extract-main} applies with $c_* = M_k^X(x_1, \dots, x_k)$, and $\bar c_N(e_N) \to c_*$ by \zcref{l:coarse-densities}. Now \eqref{e:extract-window} implies \eqref{e:coarse-moments} along every sequence, uniformly in $0 < r \le \bar r_N$. As the hypotheses hold along every sequence in the critical regime, the convergence is uniform there.
\end{proof}

\section{The coarse law}\label{s:coarse-law}

We now prove \zcref{t:coarse}. The moments of \zcref{t:coarse-moments} control the tensor powers of the mass measure away from the diagonals and the boundary. We first show that this suffices in $L^1$. For $k \ge 1$ and $\mathbf y \in (0, 1]^k$, let $j_a = \lceil N y_a \rceil$, and define the \emph{step density}
\[
 M_{k, N}(\mathbf y) = N^k \Big\langle \prod_{a = 1}^k w_\alpha(j_a) \Big\rangle_{E, r} .
\]
It depends on $E$, $r$, and $W$, which we do not display.

\begin{lemma}\label{l:coarse-L1}
Let $k \ge 1$, and let $\bar r_N \to 0$ be a deterministic sequence. Then, uniformly in the critical regime and in $0 < r \le \bar r_N$,
\begin{equation}\label{e:coarse-L1}
 \big\| M_{k, N} - M_k^{X_N(E)} \big\|_{L^1([0, 1]^k)} \longrightarrow 0 .
\end{equation}
\end{lemma}

\begin{proof}
The weights of each eigenvector sum to one, and therefore
\[
 M_{k, N} \ge 0 , \qquad \int_{[0, 1]^k} M_{k, N}(\mathbf y) \dd \mathbf y = 1 .
\]
Suppose that \eqref{e:coarse-L1} fails. Then there exist $\eta > 0$ and a sequence of parameters in the critical regime, with $r \le \bar r_N$, along which the $L^1$ distance is at least $\eta$. We pass to a subsequence along which $X_N(E)$ converges to some $X \in (0, \infty)$.

Let $\mathbf y \in (0, 1)^k$ have distinct coordinates, and choose $\delta > 0$ with
\[
 2\delta < \min_a \min(y_a, 1 - y_a) , \qquad 2\delta < \min_{a \ne b} |y_a - y_b| .
\]
For large $N$, the sites $j_a = \lceil N y_a \rceil$ satisfy the hypotheses of \zcref{t:coarse-moments} with this $\delta$. That theorem and the continuity of $M_k^X$ in \zcref{l:coarse-densities} give $M_{k, N}(\mathbf y) \to M_k^X(\mathbf y)$. The excluded set of $\mathbf y$ has Lebesgue measure zero.

We now apply Scheff\'e's lemma. The function $M_k^X$ is a probability density by \eqref{e:coarse-consistency}, and
\[
 \|M_{k, N} - M_k^X\|_{L^1} = 2 - 2 \int \min(M_{k, N}, M_k^X) .
\]
The minimum converges almost everywhere to $M_k^X$ and is bounded by it. Dominated convergence implies $\|M_{k, N} - M_k^X\|_{L^1} \to 0$. Finally, $\|M_k^{X_N(E)} - M_k^X\|_{L^1} \to 0$ by the continuity of \zcref{l:coarse-densities}. This contradicts the choice of the sequence.
\end{proof}

\begin{proof}[Proof of \zcref{t:coarse}]
Let $f \in C([0, 1]^k)$. The definitions of $M_{k, N}$ and of the cells give
\begin{align*}
 \Big\langle \int f \dd (\mu_\alpha^{(N)})^{\otimes k} \Big\rangle_{E, r}
 &= \sum_{j_1, \dots, j_k} f(j_1/N, \dots, j_k/N) \Big\langle \prod_a w_\alpha(j_a) \Big\rangle_{E, r} \\
 &= \int_{[0, 1]^k} f(\lceil N \mathbf y \rceil/N) \, M_{k, N}(\mathbf y) \dd \mathbf y .
\end{align*}
The difference between the right side and $\int f M_{k, N}$ is bounded by the modulus of continuity of $f$ at $\sqrt k/N$. Recall from \zcref{d:brownian} that $M_k^X$ is the density of the $k$-th moment measure of $\mu_X$ under $\mathbb Q_X$. Together with \zcref{l:coarse-L1}, we obtain, uniformly in the critical regime and in $0 < r \le \bar r_N$,
\begin{equation}\label{e:coarse-tensor}
 \Big\langle \int f \dd (\mu_\alpha^{(N)})^{\otimes k} \Big\rangle_{E, r} - \E_{\mathbb Q_{X_N(E)}} \int f \dd \mu_{X_N(E)}^{\otimes k} \longrightarrow 0 .
\end{equation}

Let $\mathcal A$ denote the linear span of the constant function and of the functions on $\mathcal P([0, 1])$ given by
\[
 \mu \mapsto \prod_{a = 1}^k \mu(f_a) = \int (f_1 \otimes \dots \otimes f_k) \dd \mu^{\otimes k} ,
\]
for $k \ge 1$ and $f_a \in C([0, 1])$. By \eqref{e:coarse-tensor}, the convergence holds for every $\Phi \in \mathcal A$ in place of $\int f \dd \mu^{\otimes k}$. The space $\mathcal P([0, 1])$ is compact and metrizable, and $\mathcal A$ is an algebra of continuous functions that contains the constants. It separates points, since two probability measures that agree on all continuous functions are equal. By the Stone--Weierstrass theorem, $\mathcal A$ is dense in $C(\mathcal P([0, 1]))$ for the supremum norm.

Given a continuous $\Phi$ and $\eta > 0$, we choose $\Phi_\eta \in \mathcal A$ with $\|\Phi - \Phi_\eta\|_\infty \le \eta$. Both selections in \zcref{t:coarse} are probability measures, and the difference in \zcref{t:coarse} is bounded by $2\eta$ plus the corresponding difference for $\Phi_\eta$, which tends to zero. The uniform statement follows.

For the last statement, it suffices to show that the map
\[
 X \mapsto \E_{\mathbb Q_X} \Phi(\mu_X) = \E[Z_X \Phi(\mu_X)]
\]
is continuous. We realize all $g_X$ with one Brownian motion. As $X' \to X$, we have $g_{X'} \to g_X$ uniformly on $[0, 1]$, and $Z_{X'} \to Z_X$ and $\mu_{X'} \to \mu_X$ in total variation, almost surely. For $X' \le X_+$, we have
\[
 Z_{X'} \le \exp\Big( \sqrt{2X_+} \sup_{t \le 1} |\beta_t| + X_+ \Big) ,
\]
and the continuity follows from dominated convergence, since $\Phi$ is bounded.
\end{proof}

\begin{remark}\label{r:coarse-rv}
The limit in \zcref{t:coarse} is the law found by Rifkind and Vir\'ag for critical random Schr\"odinger operators \cite[Theorem~1.1]{rifkind2018eigenvectors}. For the operator on $\{1, \dots, n\}$ whose potential has variance $\sigma^2/n$, they show that the mass of an eigenvector chosen uniformly at random converges in law to a limit that depends on a parameter $\tau$, which is determined by $\sigma$ and the energy. Let $\tau > 0$, let $\beta'$ be a two-sided standard Brownian motion and $U$ an independent uniform variable on $[0, 1]$, and set $S(t) = \exp(\beta'_t/\sqrt 2 - |t|/4)$. Their limit is the law of the random probability measure
\[
 \frac{S(\tau(t - U))}{\int_0^1 S(\tau(s - U)) \dd s} \dd t , \qquad t \in [0, 1] .
\]
For $\tau = 4X$, this is the law of $\mu_X$ under $\mathbb Q_X$. Indeed, by Brownian scaling, $(\beta'_{4Xr}/\sqrt 2)_{r \in \R}$ has the law of $(\sqrt{2X} \beta'_r)_{r \in \R}$. Hence, for each $u \in [0, 1]$, the process $(S(4X(t - u)))_{t \in [0, 1]}$ has the law of
\[
 \Big( \exp\big( \sqrt{2X} \beta'_{t - u} - X |t - u| \big) \Big)_{t \in [0, 1]} .
\]
For a standard Brownian motion $\beta$ on $[0, 1]$, the process $(\beta_t - \beta_u)_{t \in [0, 1]}$ has the law of $(\beta'_{t - u})_{t \in [0, 1]}$, since its two branches from $u$ are independent Brownian motions. Comparing with \eqref{e:intro-rooted-density}, we find that the measure above, given $U = u$, has the law of $p_{X, u}(t) \dd t$. By \zcref{l:coarse-cameron-martin}, the mixture of these laws over a uniform $U$ coincides with the law of $\mu_X$ under $\overline{\mathbb Q}_X$, and $\overline{\mathbb Q}_X$ has first marginal $\mathbb Q_X$. The same comparison identifies the pair (shape, $U$) of \cite[Theorem~1.1]{rifkind2018eigenvectors} with $(\mu_X, U)$ under $\overline{\mathbb Q}_X$, which is the limit in \zcref{t:shape}(a).
\end{remark}

\begin{proof}[Proof of \zcref{c:coarse-cov}]
The exact identities \eqref{e:intro-coarse-cov} are elementary. They follow from the definition $\dd \mathbb Q_X = Z_X \dd \Prob$ and the moment densities $M_1^X = 1$ and $M_2^X = \mathcal C_X$, by Tonelli's theorem, for bounded measurable $f$ and $g$.

For the matrix limits, let $f$ and $g$ be bounded and Riemann integrable, and set
\[
 f_N(y) = f(\lceil Ny \rceil/N) , \qquad g_N(y) = g(\lceil Ny \rceil/N) , \qquad y \in (0, 1] .
\]
As in the proof of \zcref{t:coarse},
\begin{gather*}
 \big\langle \mu_\alpha^{(N)}(f) \big\rangle_{E, r} = \int_0^1 f_N M_{1, N} \dd y , \\
 \big\langle \mu_\alpha^{(N)}(f) \, \mu_\alpha^{(N)}(g) \big\rangle_{E, r} = \int_{[0, 1]^2} f_N(y_1) g_N(y_2) M_{2, N}(y_1, y_2) \dd y_1 \dd y_2 .
\end{gather*}
The functions $f_N$ and $g_N$ are bounded by $\|f\|_\infty$ and $\|g\|_\infty$, and they converge to $f$ and $g$ at every point of continuity, hence almost everywhere. By \zcref{l:coarse-L1} at $k = 1$ and $k = 2$, we may replace $M_{1, N}$ and $M_{2, N}$ by $M_1^X = 1$ and $M_2^X = \mathcal C_X$, with an error that tends to zero. Dominated convergence then gives the limits
\[
 \int_0^1 f(y) \dd y , \qquad \iint f(y_1) g(y_2) \mathcal C_X(y_1, y_2) \dd y_1 \dd y_2 ,
\]
and the covariance converges to the right side of \eqref{e:intro-coarse-cov}.
\end{proof}

\section{Rooted laws}\label{s:coarse-rooted}

We now prove parts (a) and (b) of \zcref{t:shape}. Recall the measure $\overline{\mathbb Q}_X$ and the rooted density $p_{X, x}$ of \eqref{e:intro-rooted-density}. We first describe the law of $\mu_X$ under the tilt by $e^{g_X(x)}$, which is the law of the mass measure seen from a root at $x$.

\begin{lemma}\label{l:coarse-cameron-martin}
Let $X > 0$ and $x \in [0, 1]$. Under the probability measure $e^{g_X(x)} \dd \Prob$, the random measure $\mu_X$ has the law of $p_{X, x}(t) \dd t$ under $\Prob$. Under $\overline{\mathbb Q}_X$, the root $U$ is uniform on $[0, 1]$, and given $U = u$, the measure $\mu_X$ has the law of $p_{X, u}(t) \dd t$ under $\Prob$. The branches $(\beta_{x - t} - \beta_x)_{t \in [0, x]}$ and $(\beta_{x + t} - \beta_x)_{t \in [0, 1 - x]}$ that enter $p_{X, x}$ are independent.
\end{lemma}

\begin{proof}
The density $e^{g_X(x)} = \exp(\sqrt{2X} \beta_x - Xx)$ is the Cameron--Martin density of the drift $\sqrt{2X} \1_{[0, x]}$. By the Cameron--Martin theorem \cite{revuz1999continuous}, under $e^{g_X(x)} \dd \Prob$, the process
\[
 \hat\beta_t = \beta_t - \sqrt{2X} \, (t \wedge x)
\]
is a standard Brownian motion. In terms of $\hat\beta$, we have
\[
 g_X(t) - g_X(x) = \sqrt{2X} (\hat\beta_t - \hat\beta_x) + 2X (t \wedge x - x) - X (t - x) = \sqrt{2X} (\hat\beta_t - \hat\beta_x) - X |t - x| .
\]
The measure $\mu_X$ has the density
\[
 \frac{e^{g_X(t) - g_X(x)}}{\int_0^1 e^{g_X(s) - g_X(x)} \dd s} ,
\]
and the first statement follows. The second follows from the definition of $\overline{\mathbb Q}_X$, whose second marginal is uniform, and the third from the independence of the increments of $\hat\beta$.
\end{proof}

\begin{proof}[Proof of \zcref{t:shape}(a) and (b)]
For (a), let $\Phi$ be continuous on $\mathcal P([0, 1]) \times [0, 1]$, and set
\[
 \widetilde\Phi(\mu) = \int_0^1 \Phi(\mu, t) \, \mu(\dd t) .
\]
We first check that $\widetilde\Phi$ is continuous on $\mathcal P([0, 1])$. Let $\mu_n \to \mu$. As $\Phi$ is uniformly continuous, we have
\[
 \int \Phi(\mu_n, t) \mu_n(\dd t) - \int \Phi(\mu, t) \mu_n(\dd t) \to 0 , \qquad \int \Phi(\mu, t) \mu_n(\dd t) \to \int \Phi(\mu, t) \mu(\dd t) .
\]
Now \zcref{t:coarse} and the continuity of $X \mapsto \E_{\mathbb Q_X} \widetilde\Phi(\mu_X)$, shown in its proof, imply that the left side in (a) converges to
\[
 \E_{\mathbb Q_X} \widetilde\Phi(\mu_X) = \E \Big[ Z_X \int_0^1 \Phi(\mu_X, t) \frac{e^{g_X(t)}}{Z_X} \dd t \Big] = \E \int_0^1 \Phi(\mu_X, t) e^{g_X(t)} \dd t = \E_{\overline{\mathbb Q}_X} \Phi(\mu_X, U) ,
\]
uniformly in $0 < r \le \bar r_N$.

For (b), let $i = i_N$ be sites such that $i/N$ converges to some $x \in (0, 1)$. For $k \ge 1$, $\mathbf y \in (0, 1]^k$, and $j_a = \lceil N y_a \rceil$, we define the rooted step density as in \zcref{s:coarse-law}. By \zcref{d:selection},
\[
 M^i_{k, N}(\mathbf y) = N^k \Big\langle \prod_a w_\alpha(j_a) \Big\rangle^i_{E, r} = \frac{N^{k + 1} \langle w_\alpha(i) \prod_a w_\alpha(j_a) \rangle_{E, r}}{N \langle w_\alpha(i) \rangle_{E, r}} .
\]
Let $\mathbf y \in (0, 1)^k$ have distinct coordinates, all different from $x$. Applying \zcref{t:coarse-moments} at the orders $k + 1$ and $1$, we see that the numerator converges to $M_{k + 1}^X(x, \mathbf y)$ and the denominator to one. The rooted step densities are nonnegative with integral one, and their limit $M_{k + 1}^X(x, \cdot)$ has integral $M_1^X(x) = 1$, by \eqref{e:coarse-consistency}. As in the proof of \zcref{l:coarse-L1}, Scheff\'e's lemma implies that
\[
 \big\| M^i_{k, N} - M_{k + 1}^X(x, \cdot) \big\|_{L^1} \to 0 ,
\]
uniformly in $0 < r \le \bar r_N$.

As in the proof of \zcref{t:coarse}, we find
\[
 \Big\langle \int f \dd (\mu_\alpha^{(N)})^{\otimes k} \Big\rangle^i_{E, r} \longrightarrow \int_{[0, 1]^k} f(\mathbf y) M_{k + 1}^X(x, \mathbf y) \dd \mathbf y = \E \Big[ e^{g_X(x)} \int f \dd \mu_X^{\otimes k} \Big]
\]
for all $f \in C([0, 1]^k)$, where the last identity is the definition of $M_{k + 1}^X$. By \zcref{l:coarse-cameron-martin}, the right side is
\[
 \E \int f \dd \big( p_{X, x}(t) \dd t \big)^{\otimes k} .
\]
Part (b) now follows from the Stone--Weierstrass argument of the proof of \zcref{t:coarse}.
\end{proof}

\section{The infinite wire}\label{s:coarse-wire}

As $X \to \infty$, the mass seen from the root lives on the scale $1/X$. In the notation of \zcref{r:coarse-rv}, the limit $p_\infty(s) \dd s$ of \zcref{t:shape}(c) is the normalized measure $S(4s) \dd s$ on $\R$, by Brownian scaling. This is the shape of Dumaz and Labb\'e \cite[Theorem~3]{dumaz2023localization}, centered at the root instead of at its center of mass. We first identify the side states of the infinite wire. Recall $g_1$, $A_s$, and $\Upsilon_s$ from \zcref{s:res-scaling}, and set $A_\infty = \int_0^\infty e^{g_1(u)} \dd u$.

\begin{lemma}\label{l:coarse-side}
\begin{enumerate}[label=(\alph*)]
\item The variable $A_\infty$ is finite almost surely, and it has the law of $1/\gamma$, where $\gamma$ is a standard exponential variable.
\item For all $q > 0$, $\Upsilon_s(q)$ decreases to $\Upsilon(q) = \E e^{-q A_\infty} = 2 \sqrt q \, K_1(2 \sqrt q)$ as $s \to \infty$, and $\Upsilon_s \to \Upsilon$ in $L^2((0, \infty), \dd q)$.
\item For all $s, p > 0$, we have $\E A_s^{-p} \le s^{-p} \exp(ps/2 + p^2 s/3)$.
\end{enumerate}
\end{lemma}

\begin{proof}
The law of large numbers for $\beta$ implies that $g_1(u)/u \to -1$ almost surely, and hence $A_\infty < \infty$ almost surely. For (a), the process $\tilde\beta_u = \beta_{2u}/\sqrt 2$ is a standard Brownian motion, and
\[
 g_1(2u) = 2(\tilde\beta_u - u) , \qquad A_\infty = 2 \int_0^\infty \exp\big( 2(\tilde\beta_u - u) \big) \dd u .
\]
Dufresne's identity \cite{dufresne1990distribution}, in the form of \cite[Theorem~6.2]{matsumoto2005exponentialI}, states that for $\mu > 0$,
\[
 \int_0^\infty \exp\big( 2(\tilde\beta_u - \mu u) \big) \dd u \quad \text{has the law of} \quad \frac{1}{2\gamma_\mu} ,
\]
where $\gamma_\mu$ has the Gamma law with parameter $\mu$. With $\mu = 1$, we obtain (a).

For (b), monotone convergence and (a) give
\[
 \Upsilon_s(q) = \E e^{-q A_s} \downarrow \E e^{-q A_\infty} = \int_0^\infty e^{-q/x - x} \dd x = 2 \sqrt q \, K_1(2 \sqrt q) .
\]
The last equality is \cite[(10.32.10)]{dlmf} with $\nu = -1$ and $z = 2 \sqrt q$, together with $K_{-1} = K_1$ \cite[(10.27.3)]{dlmf}. Let $s \ge s_0 > 0$. Then $0 \le \Upsilon_s \le \Upsilon_{s_0} \le 1$, and
\[
 \int_0^\infty \Upsilon_{s_0}(q)^2 \dd q \le \int_0^\infty \Upsilon_{s_0}(q) \dd q = \E A_{s_0}^{-1} ,
\]
which is finite by (c) (with $p = 1$). The $L^2$ convergence follows by dominated convergence.

For (c), we note that Jensen's inequality implies
\[
 A_s \ge s \exp\Big( \frac1s \int_0^s g_1(u) \dd u \Big) .
\]
The variable $s^{-1} \int_0^s g_1(u) \dd u$ is Gaussian with mean $-s/2$ and variance
\[
 \frac{2}{s^2} \int_0^s \int_0^s (u \wedge v) \dd u \dd v = \frac{2s}{3} .
\]
Its moment generating function gives
\[
 \E A_s^{-p} \le s^{-p} \, \E \exp\Big( -\frac ps \int_0^s g_1(u) \dd u \Big) = s^{-p} \exp\Big( \frac{ps}{2} + \frac{p^2 s}{3} \Big) . \qedhere
\]
\end{proof}

Recall the two-sided Brownian motion $(\beta_s)_{s \in \R}$ of \eqref{e:intro-shape}. We set
\[
 g_\infty(s) = \sqrt 2 \, \beta_s - |s| , \qquad Z_\infty = \int_\R e^{g_\infty(s)} \dd s , \qquad Z_\pm = \int_0^\infty e^{g_\infty(\pm s)} \dd s ,
\]
so that $p_\infty = e^{g_\infty}/Z_\infty$ and $Z_\infty = Z_+ + Z_-$. The variables $Z_+$ and $Z_-$ are independent, and each has the law of $A_\infty$.

\begin{proof}[Proof of \zcref{t:shape}(c)]
Let $X \ge 1$ and $u \in (0, 1)$. Recall the map $\sigma_{X, u}(t) = X(t - u)$, defined before \zcref{t:shape}, and let
\[
 J_{X, u} = \sigma_{X, u}([0, 1]) = [-Xu, X(1 - u)] .
\]
Given $U = u$, \zcref{l:coarse-cameron-martin} describes $\mu_X$ under $\overline{\mathbb Q}_X$ through the standard Brownian motion $\hat\beta$ defined in the proof of that lemma, with $x = u$. The pushforward of $p_{X, u}(t) \dd t$ under $\sigma_{X, u}$ has the density
\[
 s \mapsto X^{-1} p_{X, u}(u + s/X) , \qquad s \in J_{X, u} .
\] By Brownian scaling, the process
\[
 \check\beta_s = \sqrt X \, (\hat\beta_{u + s/X} - \hat\beta_u) , \qquad s \in J_{X, u} ,
\]
is the restriction of a two-sided standard Brownian motion to $J_{X, u}$ (its two branches are independent, by \zcref{l:coarse-cameron-martin}), and
\[
 \sqrt{2X} \, (\hat\beta_{u + s/X} - \hat\beta_u) - X |s/X| = \sqrt 2 \, \check\beta_s - |s| .
\]
Consequently, the pushforward has the law of
\[
 p^{(X, u)}(s) \dd s , \qquad p^{(X, u)}(s) = \frac{e^{g_\infty(s)} \1_{J_{X, u}}(s)}{Z_{X, u}} , \qquad Z_{X, u} = \int_{J_{X, u}} e^{g_\infty} .
\]

We now realize all these laws on one probability space. Let $(\beta_s)_{s \in \R}$ be a two-sided standard Brownian motion, $g_\infty$ as above, and $U'$ an independent uniform variable on $[0, 1]$. By the preceding paragraph, the law of $((\sigma_{X, U})_* \mu_X, U)$ under $\overline{\mathbb Q}_X$ is the law of $(p^{(X, U')}(s) \dd s, U')$. As $X \to \infty$, the intervals $J_{X, U'}$ increase to $\R$ almost surely, and $Z_{X, U'} \uparrow Z_\infty$. We claim that
\begin{equation}\label{e:coarse-shape-coupling}
 \big\| p^{(X, U')} - p_\infty \big\|_{L^1(\R)} = 2 \Big( 1 - \frac{Z_{X, U'}}{Z_\infty} \Big) \longrightarrow 0 \qquad \text{almost surely} .
\end{equation}
Indeed, on $J_{X, U'}$, the difference of the densities is
\[
 e^{g_\infty} \big( Z_{X, U'}^{-1} - Z_\infty^{-1} \big) \ge 0 ,
\]
with integral $1 - Z_{X, U'}/Z_\infty$. This is also the mass of $p_\infty$ outside $J_{X, U'}$, and the $L^1$ distance is twice this value.

Convergence in $L^1$ of densities implies weak convergence of the measures. Thus
\[
 \big( p^{(X, U')}(s) \dd s, U' \big) \to \big( p_\infty(s) \dd s, U' \big) \qquad \text{almost surely in } \mathcal P(\R) \times [0, 1] ,
\]
and the laws converge weakly. The same argument with $u = x$ fixed proves the second statement.
\end{proof}

\section{The mean shape}\label{s:coarse-mean}

We now prove \zcref{c:mean-shape}, and we show that $X^{-1} \mathcal C_X(x, x) \to \bar p_\infty(0)$, as announced after \zcref{c:mean-shape}. The first step writes the mean shape in terms of the operator $\mathscr L$ of \zcref{l:res-limit-semigroups}.

\begin{lemma}\label{l:coarse-mean-pairing}
For all $s \ge 0$,
\begin{equation}\label{e:coarse-mean-pairing}
 \bar p_\infty(s) = \int_0^\infty \Upsilon(q) \big( e^{-s \mathscr L} \Upsilon \big)(q) \dd q = \big\langle \Upsilon, e^{-s \mathscr L} \Upsilon \big\rangle_{L^2((0, \infty), \dd q)} .
\end{equation}
In particular, $\bar p_\infty(0) = \int_0^\infty \Upsilon(q)^2 \dd q$. Moreover, for $x \in (0, 1)$, we have $X^{-1} \mathcal C_X(x, x) \downarrow \bar p_\infty(0)$ as $X \to \infty$.
\end{lemma}

\begin{proof}
By the symmetry $\beta \mapsto (\beta_{-s})_{s \in \R}$, it suffices to consider $s \ge 0$. We insert
\[
 \frac{1}{Z_\infty} = \int_0^\infty e^{-q Z_\infty} \dd q ,
\]
and we use the independence of $Z_+$ and $Z_-$ and Tonelli's theorem. We find
\[
 \bar p_\infty(s) = \int_0^\infty \E e^{-q Z_-} \, \E \big[ e^{g_\infty(s)} e^{-q Z_+} \big] \dd q .
\]

Since $Z_-$ has the law of $A_\infty$, \zcref{l:coarse-side}(b) gives $\E e^{-q Z_-} = \Upsilon(q)$. For the second factor, we split
\[
 Z_+ = \int_0^s e^{g_\infty} + e^{g_\infty(s)} Z'_s , \qquad Z'_s = \int_0^\infty e^{g_\infty(s + v) - g_\infty(s)} \dd v ,
\]
where $Z'_s$ is independent of $(\beta_v)_{0 \le v \le s}$ and has the law of $A_\infty$ (by the Markov property of $\beta$ at time $s$). We have $g_\infty = g_1$ on $[0, \infty)$, with $g_1$ built from $\beta$. Conditioning on $(\beta_v)_{0 \le v \le s}$ and using \zcref{l:coarse-side}(b), we obtain
\[
 \E \big[ e^{g_\infty(s)} e^{-q Z_+} \big] = \E \big[ e^{g_1(s)} e^{-q A_s} \Upsilon(q e^{g_1(s)}) \big] = q^{-1} \mathcal T_s(q \Upsilon)(q) = \big( e^{-s \mathscr L} \Upsilon \big)(q) ,
\]
where the last step is \eqref{e:res-h-transform}. The second identity in \eqref{e:coarse-mean-pairing} follows from \zcref{l:res-limit-semigroups}(c), since $\Upsilon \in L^2$ by \zcref{l:coarse-side}(b).

For the last statement, \zcref{l:coarse-cameron-martin} and the definition of $\mathcal C_X$ give
\[
 \mathcal C_X(x, x) = \E \Big[ e^{g_X(x)} \frac{e^{g_X(x)}}{Z_X} \Big] = \E \, p_{X, x}(x) = X \, \E \big[ Z_{X, x}^{-1} \big] ,
\]
where $Z_{X, x}$ is the normalization of $p^{(X, x)}$ in the proof of \zcref{t:shape}(c), by the scaling used there. As $X$ increases, $J_{X, x}$ increases to $\R$, and $Z_{X, x}^{-1}$ decreases to $Z_\infty^{-1}$. For $X \ge 1/\min(x, 1 - x)$, the interval $J_{X, x}$ contains $[-1, 1]$, and
\[
 \E Z_{X, x}^{-1} \le \E \Big( \int_{-1}^1 e^{g_\infty} \Big)^{-1} < \infty ,
\]
by Jensen's inequality, as in \zcref{l:coarse-side}(c). Monotone convergence yields
\[
 X^{-1} \mathcal C_X(x, x) = \E Z_{X, x}^{-1} \downarrow \E Z_\infty^{-1} = \bar p_\infty(0) . \qedhere
\]
\end{proof}

To compute the right side of \eqref{e:coarse-mean-pairing}, we diagonalize $\mathscr L$. By \zcref{l:res-limit-semigroups}(b), the map $\mathcal U$ conjugates $\mathscr L$ to the closure $\mathscr H$ of $-\partial_y^2 + e^y + 1/4$ on $C_c^\infty(\R)$. For $k > 0$ and $y \in \R$, we set
\[
 \psi_k(y) = \frac 2\pi \big( k \sinh(2\pi k) \big)^{1/2} K_{2 \iu k}(2 e^{y/2}) .
\]
The function $K_{\iu \tau}(x)$ is real for $x, \tau > 0$, and $\psi_k$ is real. It is bounded, since $K_{\iu \tau}(x)$ is bounded as $x \downarrow 0$ and decays exponentially as $x \to \infty$.

\begin{lemma}\label{l:coarse-kl}
There exists an isometry $\mathcal W \colon L^2(\R, \dd y) \to L^2((0, \infty), \dd k)$ such that $(\mathcal W v)(k) = \int_\R \psi_k(y) v(y) \dd y$ for almost every $k$, for all $v \in L^1(\R) \cap L^2(\R)$, and
\begin{equation}\label{e:coarse-kl}
 \big\langle v, e^{-s \mathscr H} v \big\rangle_{L^2(\R)} = \int_0^\infty e^{-s (k^2 + 1/4)} |(\mathcal W v)(k)|^2 \dd k , \qquad v \in L^2(\R), \ s \ge 0 .
\end{equation}
\end{lemma}

\begin{proof}
The lemma is the Kontorovich--Lebedev inversion \cite[Section 10.43(v)]{dlmf}, written in the variables of $\mathscr H$. The normalization of $\psi_k$ enters the constant of \zcref{c:mean-shape}, and we first check it. Let $\omega$ be a function on $(0, \infty)$ such that $x^{-1} \omega(x)$ is continuously differentiable on $(0, \infty)$, and the functions $x \omega(x)$ and $x \, \dd(x^{-1} \omega(x))/\dd x$ are absolutely integrable on $(0, \infty)$; this is condition (a) of \cite[Section 10.43(v)]{dlmf}. Then \cite[(10.43.30)--(10.43.31)]{dlmf} state that
\begin{equation}\label{e:coarse-kl-dlmf}
 \omega(x) = \int_0^\infty \Omega(\tau) K_{\iu \tau}(x) \dd \tau , \qquad
 \Omega(\tau) = \frac{2 \tau \sinh(\pi \tau)}{\pi^2} \int_0^\infty \omega(x) K_{\iu \tau}(x) \frac{\dd x}{x} .
\end{equation}

Let $v \in C_c^\infty(\R)$, and set
\[
 (\mathcal W v)(k) = \int_\R \psi_k(y) v(y) \dd y , \qquad k > 0 .
\] The function $\omega(x) = v(2 \log(x/2))$ is smooth with compact support in $(0, \infty)$ and satisfies condition (a). We substitute $x = 2 e^{y/2}$, for which $\dd x/x = \dd y/2$, and $\tau = 2k$. By the definition of $\psi_k$, the second identity in \eqref{e:coarse-kl-dlmf} becomes
\[
 \Omega(2k) = \frac{2k \sinh(2\pi k)}{\pi^2} \int_\R v(y) K_{2 \iu k}(2 e^{y/2}) \dd y = \frac{(k \sinh(2\pi k))^{1/2}}{\pi} \, (\mathcal W v)(k) ,
\]
and the first identity, at $x = 2 e^{y/2}$, becomes
\begin{equation}\label{e:coarse-kl-inversion}
 v(y) = 2 \int_0^\infty \Omega(2k) K_{2 \iu k}(2 e^{y/2}) \dd k = \int_0^\infty \psi_k(y) (\mathcal W v)(k) \dd k .
\end{equation}
Thus the factor $\frac2\pi (k \sinh(2\pi k))^{1/2}$ in $\psi_k$ is the one for which the transform $\mathcal W$ and the expansion \eqref{e:coarse-kl-inversion} are inverse to each other.

Next, we deduce the isometry. Since \cite[Section 10.43(v)]{dlmf} does not state it, we include the short argument. We multiply \eqref{e:coarse-kl-inversion} by $\overline{v(y)}$ and integrate over $y$, and for the interchange of the integrals we use two bounds.

First, by \cite[(10.32.9)]{dlmf}, we have
\[
 K_{\iu \tau}(x) = \frac12 \int_\R e^{-x \cosh t + \iu \tau t} \dd t , \qquad x > 0 , \ \tau \ge 0 .
\] We shift the contour to $\R + \iu \theta$ with $0 \le \theta < \pi/2$, which is allowed since $|e^{-x \cosh(t + \iu \theta)}| = e^{-x \cos\theta \cosh t}$, and we arrive at
\begin{equation}\label{e:coarse-K-bound}
 |K_{\iu \tau}(x)| \le e^{-\theta \tau} K_0(x \cos\theta) , \qquad 0 \le \theta < \pi/2 .
\end{equation}
With $\tau = 2k$ and $\theta = \pi/2 - (1 + 2k)^{-1}$, this implies that
\[
 |\psi_k(y)| \le C (1 + k)^{1/2} \big( 1 + \log(1 + k) \big) ,
\]
uniformly for $y$ in compact sets.

Second, the modified Bessel equation $(x \partial_x)^2 K_{\iu \tau} = (x^2 - \tau^2) K_{\iu \tau}$ of \cite[(10.25.1)]{dlmf} becomes $(-\partial_y^2 + e^y) \psi_k = k^2 \psi_k$, since $x \partial_x = 2 \partial_y$, and two integrations by parts give
\begin{equation}\label{e:coarse-kl-eigen}
 \mathcal W (\mathscr H v) = (k^2 + 1/4) \, \mathcal W v , \qquad v \in C_c^\infty(\R) .
\end{equation}
Applying \eqref{e:coarse-kl-eigen} $n$ times and using the first bound, we find $|(\mathcal W v)(k)| \le C_n (1 + k)^{-n}$ for all $n$. Hence the double integral converges absolutely, and since $\psi_k$ is real, we obtain
\[
 \int_\R |v(y)|^2 \dd y = \int_0^\infty |(\mathcal W v)(k)|^2 \dd k , \qquad v \in C_c^\infty(\R) .
\]

Therefore $\mathcal W$ extends from $C_c^\infty(\R)$ to an isometry. For $v \in L^1 \cap L^2$, we approximate $v$ by $v_n \in C_c^\infty$ in $L^1$ and $L^2$. As $\psi_k$ is bounded, the integrals of $\psi_k v_n$ converge to that of $\psi_k v$ for every $k$, and a subsequence of $\mathcal W v_n$ converges almost everywhere to $\mathcal W v$.

It remains to prove \eqref{e:coarse-kl}. Let $\mathsf M$ denote multiplication by $k^2 + 1/4$. By \eqref{e:coarse-kl-eigen}, we have
\[
 \mathcal W (\mathscr H \pm \iu)^{-1} w = (\mathsf M \pm \iu)^{-1} \mathcal W w , \qquad w \in (\mathscr H \pm \iu) C_c^\infty(\R) .
\] This set is dense, since $\mathscr H$ is essentially self-adjoint on $C_c^\infty(\R)$ by \zcref{l:res-limit-semigroups}(b). By continuity, the identity holds for all $w \in L^2$.

By the Stone--Weierstrass theorem, the polynomials in $(\lambda + \iu)^{-1}$ and $(\lambda - \iu)^{-1}$ are dense in the continuous functions on $[1/4, \infty]$. Functional calculus implies that $\mathcal W e^{-s \mathscr H} = e^{-s \mathsf M} \mathcal W$, and since $\mathcal W$ is an isometry, we conclude that
\[
 \big\langle v, e^{-s \mathscr H} v \big\rangle = \big\langle \mathcal W v, e^{-s \mathsf M} \mathcal W v \big\rangle ,
\]
which is \eqref{e:coarse-kl}.
\end{proof}

\begin{proof}[Proof of \zcref{c:mean-shape}]
The function $\bar p_\infty$ is even, by the symmetry $\beta \mapsto (\beta_{-s})_{s \in \R}$ of the two-sided Brownian motion, and Tonelli's theorem gives
\[
 \int_\R \bar p_\infty(s) \dd s = \E \int_\R p_\infty(s) \dd s = 1 .
\]

We next compute $\bar p_\infty(0)$, by Beta--Gamma algebra. By \zcref{l:coarse-side}(a), the variables $\gamma_\pm = 1/Z_\pm$ are independent standard exponential variables, and
\[
 \bar p_\infty(0) = \E \frac{1}{Z_+ + Z_-} = \E \frac{\gamma_+ \gamma_-}{\gamma_+ + \gamma_-} .
\]
The sum $\gamma_+ + \gamma_-$ has the Gamma law with parameter two, the ratio $\gamma_+/(\gamma_+ + \gamma_-)$ is uniform on $[0, 1]$, and the two are independent. We write
\[
 \frac{\gamma_+ \gamma_-}{\gamma_+ + \gamma_-} = (\gamma_+ + \gamma_-) \cdot \frac{\gamma_+}{\gamma_+ + \gamma_-} \cdot \frac{\gamma_-}{\gamma_+ + \gamma_-} ,
\]
and we obtain
\[
 \bar p_\infty(0) = \E[\gamma_+ + \gamma_-] \int_0^1 v (1 - v) \dd v = 2 \cdot \frac16 = \frac13 .
\]

For the tail, we apply \zcref{l:coarse-mean-pairing}, the identity
\[
 e^{-s \mathscr L} = \mathcal U^{-1} e^{-s \mathscr H} \mathcal U ,
\]
and \zcref{l:coarse-kl} with $v = \mathcal U \Upsilon$. The function $\mathcal U \Upsilon(y) = e^{y/2} \Upsilon(e^y)$ is integrable and square integrable, since $\Upsilon \le \Upsilon_1 \le \bar\Upsilon_{1, 1}$ by \zcref{l:res-envelope}. We obtain
\begin{equation}\label{e:coarse-spectral}
 \bar p_\infty(s) = e^{-s/4} \int_0^\infty e^{-s k^2} \varpi(k) \dd k , \qquad \varpi(k) = \Big| \int_\R \psi_k(y) e^{y/2} \Upsilon(e^y) \dd y \Big|^2 .
\end{equation}

We now compute $\varpi$. With $x = 2 e^{y/2}$, we have
\[
 e^{y/2} \Upsilon(e^y) = 2 e^y K_1(x) = \frac{x^2}{2} K_1(x) , \qquad \dd y = \frac{2 \dd x}{x} ,
\]
and therefore
\[
 \int_\R \psi_k(y) e^{y/2} \Upsilon(e^y) \dd y = \frac 2\pi \big( k \sinh(2\pi k) \big)^{1/2} \int_0^\infty x K_1(x) K_{2 \iu k}(x) \dd x .
\]

We evaluate the last integral with three identities. The product formula \cite[(10.32.17)]{dlmf} gives
\[
 K_1(x) K_{2 \iu k}(x) = 2 \int_0^\infty K_{1 + 2 \iu k}(2x \cosh t) \cosh\big( (1 - 2\iu k) t \big) \dd t .
\]

Next, \cite[(10.43.19)]{dlmf} with $\mu = 2$ and $\nu = 1 + 2 \iu k$, and the substitution $x' = 2x \cosh t$, give
\[
 \int_0^\infty x K_{1 + 2\iu k}(2x \cosh t) \dd x = (2 \cosh t)^{-2} \, \Gamma(\tfrac12 - \iu k) \Gamma(\tfrac32 + \iu k) .
\]
The remaining integral is a beta integral. With $w' = e^{2t}$,
\[
 \int_0^\infty \frac{\cosh(a t)}{\cosh^2 t} \dd t = \int_0^\infty \frac{(w')^{a/2}}{(1 + w')^2} \dd w' = \Gamma(1 + \tfrac a2) \Gamma(1 - \tfrac a2) , \qquad |\Re a| < 2 ,
\]
which we use with $a = 1 - 2\iu k$. We combine these three identities with
\[
 \big| \Gamma(\tfrac12 + \iu k) \big|^2 = \frac{\pi}{\cosh(\pi k)} , \qquad \Gamma(\tfrac32 + \iu k) = (\tfrac12 + \iu k) \Gamma(\tfrac12 + \iu k) ,
\]
where the first identity is \cite[(5.4.4)]{dlmf}. We find
\[
 \int_0^\infty x K_1(x) K_{2 \iu k}(x) \dd x = \frac12 \big| \Gamma(\tfrac12 + \iu k) \big|^2 \big| \Gamma(\tfrac32 + \iu k) \big|^2 = \frac{\pi^2 (k^2 + 1/4)}{2 \cosh^2(\pi k)} .
\]
Inserting this into the definition of $\varpi$, we obtain
\begin{equation}\label{e:coarse-weight}
 \varpi(k) = 2\pi^2 k \Big( k^2 + \frac14 \Big)^2 \frac{\tanh(\pi k)}{\cosh^2(\pi k)} .
\end{equation}
Setting $s = 0$ in \eqref{e:coarse-spectral} and using $\bar p_\infty(0) = 1/3$, we find
\[
 \int_0^\infty \varpi(k) \dd k = \frac13 .
\]
A numerical evaluation of this integral of \eqref{e:coarse-weight} gives the same value, which confirms the normalization of \zcref{l:coarse-kl}.

Finally, \eqref{e:coarse-weight} implies that
\[
 \varpi(k) = \frac{\pi^3}{8} k^2 + O(k^4) \quad \text{as } k \downarrow 0 , \qquad \varpi(k) \le C k^5 e^{-\pi k} \quad \text{for } k \ge 1 .
\]
We split the integral in \eqref{e:coarse-spectral} at $k = 1$ and use
\[
 \int_0^\infty k^2 e^{-sk^2} \dd k = \frac{\sqrt\pi}{4} s^{-3/2} , \qquad \int_0^\infty k^4 e^{-sk^2} \dd k = \frac{3\sqrt\pi}{8} s^{-5/2} .
\]
We obtain
\[
 \int_0^\infty e^{-sk^2} \varpi(k) \dd k = \frac{\pi^{7/2}}{32 \, s^{3/2}} \big( 1 + O(s^{-1}) \big) , \qquad s \to \infty ,
\]
which proves \eqref{e:intro-tail}.
\end{proof}

The second moment of the mean shape is also explicit.

\begin{lemma}\label{l:coarse-second-moment}
We have $\int_\R s^2 \bar p_\infty(s) \dd s = 8\zeta(3)$.
\end{lemma}

\begin{proof}
By the evenness of $\bar p_\infty$, \eqref{e:coarse-spectral} and Tonelli's theorem,
\[
 \int_\R s^2 \bar p_\infty(s) \dd s = 2 \int_0^\infty \varpi(k) \int_0^\infty s^2 e^{-s(k^2 + 1/4)} \dd s \dd k = 4 \int_0^\infty \frac{\varpi(k)}{(k^2 + 1/4)^3} \dd k = 8\pi^2 J ,
\]
where we used \eqref{e:coarse-weight} and set
\[
 J = \int_0^\infty \frac{k \tanh(\pi k)}{\cosh^2(\pi k) (k^2 + 1/4)} \dd k .
\]
It remains to show that $J = \zeta(3)/\pi^2$, and we do this by a residue computation. We use the identity
\[
 \tanh(\pi k) \operatorname{sech}^2(\pi k) = -\frac{1}{2\pi} \, \partial_k \operatorname{sech}^2(\pi k) ,
\]
where $\operatorname{sech} = 1/\cosh$. An integration by parts without boundary terms gives
\[
 J = \frac{1}{2\pi} \int_0^\infty \operatorname{sech}^2(\pi k) \, \frac{1/4 - k^2}{(k^2 + 1/4)^2} \dd k = \frac{1}{2\pi} \Re \int_0^\infty f(k) \dd k , \qquad f(k) = \frac{\operatorname{sech}^2(\pi k)}{(1/2 + \iu k)^2} .
\]
For real $k$, we have $f(-k) = \overline{f(k)}$, and therefore
\[
 J = \frac{1}{4\pi} \int_\R f(k) \dd k .
\]

We now close the contour in the lower half-plane. Let $n \ge 1$ be an integer and $R > 0$, and consider the rectangle with vertices $\pm R$ and $\pm R - \iu n$. On its bottom side, we have
\[
 |\cosh(\pi(x - \iu n))| = \cosh(\pi x) , \qquad |1/2 + \iu k| \ge n ,
\]
and this side contributes $O(n^{-2})$. On the vertical sides, $|\cosh(\pi k)| \ge \sinh(\pi R)$, and they contribute $O(n e^{-2\pi R})$.

Inside the rectangle, the poles of $f$ are the zeros $k_m = -\iu (m + 1/2)$, $0 \le m < n$, of $\cosh(\pi k)$. Near $k_m$, we have $\cosh(\pi k) = -\iu (-1)^m \sinh(\pi (k - k_m))$, and hence
\[
 \operatorname{sech}^2(\pi k) = -\frac{1}{\pi^2 (k - k_m)^2} + O(1) ,
\]
with no term of order $(k - k_m)^{-1}$, since $\sinh^2$ is even. We write $(1/2 + \iu k)^2 = -(k - \iu/2)^2$. The residue of $f$ at $k_m$ is therefore
\[
 \frac{1}{\pi^2} \, \partial_k (k - \iu/2)^{-2} \big|_{k = k_m} = -\frac{2}{\pi^2} \big( -\iu (m + 1) \big)^{-3} = \frac{2\iu}{\pi^2 (m + 1)^3} .
\]
The rectangle is traversed clockwise, and we let $R \to \infty$ and then $n \to \infty$. We obtain
\[
 \int_\R f(k) \dd k = -2\pi\iu \sum_{m \ge 0} \frac{2\iu}{\pi^2 (m + 1)^3} = \frac{4}{\pi} \, \zeta(3) ,
\]
and we conclude that
\[
 J = \frac{1}{4\pi} \cdot \frac{4\zeta(3)}{\pi} = \frac{\zeta(3)}{\pi^2} . \qedhere
\]
\end{proof}

\begin{remark}\label{r:coarse-mean-physics}
The tail \eqref{e:intro-tail}, with the same constant, is the asymptotic form of the eigenfunction correlation of an infinite wire obtained by Efetov and Larkin \cite[(56)]{efetov1983kinetics}, in units in which their localization length $L_c$ is one. Ivanov, Ostrovsky, and Skvortsov \cite[(75)]{ivanov2009correlations} give the zero-frequency limit $R(0, x)$ of the correlation of the local density of states of an infinite wire, which describes the decay of a single eigenfunction, in the form
\[
 R(0, x) = 4\pi^2 \partial_t^2 \int_0^\infty k \tanh(\pi k) \cosh^{-2}(\pi k) e^{-(k^2 + 1/4) t} \dd k ,
\]
where $t = x/\xi$. By \eqref{e:coarse-spectral} and \eqref{e:coarse-weight}, we have $R(0, x) = 2\bar p_\infty(t)$, where the factor two is their normalization of the correlator. The expansion of Efetov and Larkin \cite[(57)]{efetov1983kinetics} corresponds to the second moment $\int_\R s^2 \bar p_\infty(s) \dd s = 8\zeta(3)$, which is \zcref{l:coarse-second-moment}. (The supplementary script \texttt{coarse\_constants.py} confirms this value, and the tail constant, numerically.)
\end{remark}

\section{Endpoints of the coarse law}\label{s:coarse-endpoints}

We now prove the third and fourth rows of the table in \zcref{p:endpoints}.

\begin{proposition}\label{p:coarse-endpoints}
As $X \downarrow 0$, the law of $\mu_X$ under $\mathbb Q_X$ converges weakly to the point mass $\delta_{\mathrm{Leb}}$, and $\mathcal C_X(x, y) \to 1$ uniformly in $x, y \in [0, 1]$. As $X \to \infty$, the law of $\mu_X$ under $\mathbb Q_X$ converges weakly to the law of $\delta_U$, with $U$ uniform on $[0, 1]$, and for $0 < x < y < 1$, there exists $C$ such that
\begin{equation}\label{e:coarse-correlation-decay}
 \mathcal C_X(x, y) \le C X e^{-X (y - x)/4} , \qquad X \ge 1/\min(x, 1 - y) .
\end{equation}
\end{proposition}

\begin{proof}
For $X \downarrow 0$, we realize all $g_X$ with one Brownian motion. Then $g_X \to 0$ uniformly on $[0, 1]$, and $Z_X \to 1$ and the density of $\mu_X$ converges uniformly to one, almost surely. For $X \le 1$, the variables $Z_X$ and the integrands in the definition of $M_2^X$ are bounded by
\[
 \exp\Big( 3 \big( \sqrt 2 \sup_{t \le 1} |\beta_t| + 1 \big) \Big) ,
\]
which is integrable. Dominated convergence yields
\[
 \E[Z_X \Phi(\mu_X)] \to \Phi(\mathrm{Leb})
\]
for every continuous $\Phi$ on $\mathcal P([0, 1])$, and $\mathcal C_X \to 1$ uniformly.

For $X \to \infty$, we use the coupling in the proof of \zcref{t:shape}(c). Under $\overline{\mathbb Q}_X$, the pair $(\mu_X, U)$ has the law of $(\mu^{(X)}, U')$, where $\mu^{(X)}$ is the pushforward of $p^{(X, U')}(s) \dd s$ under $s \mapsto U' + s/X$. Let $\eta > 0$. The Wasserstein distance satisfies
\begin{align*}
 W_1(\mu^{(X)}, \delta_{U'}) = \int |t - U'| \, \mu^{(X)}(\dd t) &\le \eta + \int_{|s| > X\eta} p^{(X, U')}(s) \dd s \\
 &\le \eta + \big\| p^{(X, U')} - p_\infty \big\|_{L^1} + \int_{|s| > X\eta} p_\infty(s) \dd s .
\end{align*}
By \eqref{e:coarse-shape-coupling}, the right side tends to $\eta$ almost surely, and $W_1(\mu^{(X)}, \delta_{U'}) \to 0$ almost surely. Since $\mathbb Q_X$ is the first marginal of $\overline{\mathbb Q}_X$, the law of $\mu_X$ under $\mathbb Q_X$ converges weakly to the law of $\delta_{U'}$.

For \eqref{e:coarse-correlation-decay}, \zcref{l:coarse-cameron-martin} and the scaling of the proof of \zcref{t:shape}(c) give
\[
 \mathcal C_X(x, y) = \E \, p_{X, x}(y) = X \, \E \Big[ \frac{e^{g_\infty(s)}}{Z_{X, x}} \Big] , \qquad s = X(y - x) .
\]
As $X \ge 1/\min(x, 1 - y)$, the interval $J_{X, x}$ contains $[-1, 0]$ and $[s, s + 1]$. Let
\[
 Z^{(0)} = \int_{-1}^0 e^{g_\infty(v)} \dd v , \qquad Z^{(s)} = \int_0^1 e^{g_\infty(s + v) - g_\infty(s)} \dd v .
\]
Then
\[
 Z_{X, x} \ge Z^{(0)} + e^{g_\infty(s)} Z^{(s)} , \qquad
 \frac{e^{g_\infty(s)}}{Z_{X, x}} \le \min\Big( \frac{e^{g_\infty(s)}}{Z^{(0)}}, \frac{1}{Z^{(s)}} \Big) \le \frac{e^{g_\infty(s)/2}}{(Z^{(0)} Z^{(s)})^{1/2}} .
\]
The variables $g_\infty(s)$, $Z^{(0)}$, and $Z^{(s)}$ depend on the increments of $\beta$ on $[0, s]$, $[-1, 0]$, and $[s, s + 1]$, and they are independent. The exponent $g_\infty(s)/2$ is Gaussian with mean $-s/2$ and variance $s/2$, and
\[
 \E e^{g_\infty(s)/2} = \E e^{\beta_s/\sqrt 2 - s/2} = e^{-s/4} ,
\]
and the variables $Z^{(0)}$ and $Z^{(s)}$ have the same law, with $\E (Z^{(0)})^{-1/2} < \infty$ by Jensen's inequality, as in \zcref{l:coarse-side}(c). Together, these bounds give \eqref{e:coarse-correlation-decay}.
\end{proof}