\chapter{The limiting point process}\label{s:process}

In this section, we prove \zcref{t:process}. Parts (a) and (b) concern the unfolded spectral process $\Xi_N$: its convergence in law to a unique point process $\Xi_X$, the source law \eqref{e:intro-source-law}, and the stationarity, reflection invariance, simplicity, and unit intensity of $\Xi_X$. In part (c), we compute the first two derivatives of the rank-one source. The inputs are the matrix bounds of \zcref{s:interface}, part (a) of \zcref{t:interface}, and the partition function of \zcref{s:sigma}, with its derivatives at rank one from \zcref{l:sigma-jets}.

The proof has three steps. First, the factorial moment bound of \zcref{l:iface-counts} implies tightness and shows that every subsequential limit is simple and of unit intensity. The scalar Gaussian component of $H$ in \zcref{l:iface-gaussian} shows that every subsequential limit is stationary. Second, we show that the expectations of the products $\prod_{x \in \Xi} R(x)$ over the rational functions $R$ of \eqref{e:intro-source-law}, which all have the denominator $(x^2 + \eps^2)^k$ for one fixed $\eps$, determine the law of a point process. After the Cayley map $x = \eps \tan(\vartheta/2)$, the differences $R - 1$ become trigonometric polynomials, and we obtain the required approximation of Laplace functionals from Fej\'er's theorem. This is the main step specific to our setting. Third, \zcref{t:interface}(a) identifies $T_N^{(k)}$ only near the base point of the source parameters. Since $T_N^{(k)}$ is entire in the fermionic source parameters, with bounds uniform in $N$, a normal-families argument extends the convergence to all parameters.

In \zcref{s:proc-jets}, we identify the first rank-one derivative with the unit intensity of $\Xi_X$, and we take the second from \zcref{l:sigma-jets}, which computes it in the sigma model by a Dyson expansion and supersymmetric localization. The function $I_X$ comes from the two pairings in this expansion with one retarded and one advanced insertion, to which localization does not apply; the radial integration formula of \zcref{l:sigma-radial-integral} writes each of them as its corner term $-1$ minus the kernel $K_X(x, y; 2 \pi \eps)$, and integrating this kernel over the positions $0 < x < y < 1$ of the two insertions gives $I_X(2 \pi \eps)$. Besides this lemma, only the analyticity of the partition function, which \zcref{l:proc-entire} takes from \zcref{l:sigma-entire}, uses the structure of the sigma model.

Throughout this section, $(N, W, E)$ tends to the critical limit at a fixed $X \in (0, \infty)$, in the sense of \zcref{d:intro-critical}, and $\eps > 0$ is fixed. We let $\mathcal N(\R)$ denote the space of locally finite counting measures on $\R$ with the vague topology of \zcref{s:intro-results}, which is a Polish space \cite[Chapter~9]{daley2008introduction}. A random point measure is a random element of $\mathcal N(\R)$; multiple points are allowed. For $\zeta \in \mathcal N(\R)$ and a function $f$, we write $\zeta(f) = \int f \dd \zeta$. For a function $R \colon \R \to \C$ with $\sum_{x \in \zeta} |R(x) - 1| < \infty$, we let $\prod_{x \in \zeta} R(x)$ denote the absolutely convergent product over the points of $\zeta$, counted with multiplicity. When $R > 0$, this product equals $\exp \zeta(\log R)$. Recall that $P_\eps(x) = \eps/(\pi (x^2 + \eps^2))$ denotes the Poisson kernel.

\section{Ratios with poles at \texorpdfstring{$\pm \iu \eps$}{+-i epsilon}}\label{s:proc-ratios}

For $k \ge 1$, $\mathbf u \in \R^k$ with $\sum_j u_j = 0$ and $\mathbf b \in (0, \infty)^k$, we define
\begin{equation}\label{e:proc-ratio}
 R_{k, \mathbf u, \mathbf b}(x) = \frac{\prod_{j = 1}^k \bigl( (x - u_j)^2 + b_j^2 \bigr)}{(x^2 + \eps^2)^k}, \qquad x \in \R .
\end{equation}
By \eqref{e:intro-ratio}, we have
\[
 T_N^{(k)}(\mathbf u, \mathbf b) = \E \prod_{x \in \Xi_N} R_{k, \mathbf u, \mathbf b}(x) .
\]
The centers $u_j$ and the heights $b_j$ of the numerator vary independently, subject to $\sum_j u_j = 0$. The denominator $(x^2 + \eps^2)^k$ depends on $k$, but it involves the one damping $\eps$ for all $k$, so that every ratio $R_{k, \mathbf u, \mathbf b}$ has its poles at $\pm \iu \eps$. We also define $\mathcal V$ to be the set of functions
\begin{equation}\label{e:proc-algebra}
 x \mapsto a(x) P_\eps(x)^k, \qquad k \ge 1, \quad a \in \R[x], \quad \deg a \le 2k - 2 .
\end{equation}

\begin{lemma}\label{l:proc-algebra}
The following statements hold.
\begin{enumerate}[label=(\alph*)]
\item The set $\mathcal V$ is a real algebra of functions, and it is spanned by the functions $P_\eps^j$ with $j \ge 1$ and $x P_\eps^j$ with $j \ge 2$.
\item Under the Cayley map $x = \eps \tan(\vartheta/2)$, $\vartheta \in (-\pi, \pi)$, which is the stereographic parametrization of $\R \cup \{\infty\}$ by the circle, we have
\begin{equation}\label{e:proc-cayley}
 P_\eps(x) = \frac{1 + \cos \vartheta}{2 \pi \eps}, \qquad x P_\eps(x) = \frac{\sin \vartheta}{2 \pi} ,
\end{equation}
and the set $P_\eps^{-1} \mathcal V = \{ f/P_\eps : f \in \mathcal V \}$ is the algebra of real trigonometric polynomials in $\vartheta$.
\item For all $k$, $\mathbf u$, and $\mathbf b$ as in \eqref{e:proc-ratio}, we have $R_{k, \mathbf u, \mathbf b} - 1 \in \mathcal V$. Moreover, $R_{k, \mathbf u, \mathbf b}$ is bounded below by a positive constant, and there exists $C_R > 0$ such that $|\log R_{k, \mathbf u, \mathbf b}(x)| \le C_R P_\eps(x)$ for all $x \in \R$.
\end{enumerate}
\end{lemma}

\begin{proof}
We begin with part (a). Since $(x^2 + \eps^2)^{-1} = (\pi/\eps) P_\eps(x)$, we have, for $j \ge k$,
\[
 a P_\eps^k = \Bigl( \frac \pi \eps \Bigr)^{j - k} a \, (x^2 + \eps^2)^{j - k} P_\eps^j ,
\]
and the polynomial $a (x^2 + \eps^2)^{j - k}$ has degree at most $2j - 2$ if $\deg a \le 2k - 2$. We may thus write two elements of $\mathcal V$ with the same power of $P_\eps$, and $\mathcal V$ is closed under sums. The product of $a_1 P_\eps^{k_1}$ and $a_2 P_\eps^{k_2}$ has a numerator of degree at most $2(k_1 + k_2) - 4$, and $\mathcal V$ is closed under products.

For the spanning set, let $c \in \{0, 1\}$ and $2i + c \le 2k - 2$. Expanding $x^{2i} = (x^2 + \eps^2 - \eps^2)^i$ and using $(x^2 + \eps^2) P_\eps = \eps/\pi$, we find
\[
 x^{2i + c} P_\eps^k = x^c \sum_{i' = 0}^{i} \binom{i}{i'} (-\eps^2)^{i - i'} \Bigl( \frac \eps \pi \Bigr)^{i'} P_\eps^{k - i'} .
\]
The condition $2i + c \le 2k - 2$ implies that $k - i' \ge k - i \ge 1$ for $c = 0$ and $k - i' \ge 2$ for $c = 1$. Conversely, the functions $P_\eps^j$ with $j \ge 1$ and $x P_\eps^j$ with $j \ge 2$ belong to $\mathcal V$.

For part (b), the identity $1 + \tan^2(\vartheta/2) = \cos^{-2}(\vartheta/2)$ gives
\[
 P_\eps(x) = \frac{\cos^2(\vartheta/2)}{\pi \eps} = \frac{1 + \cos \vartheta}{2 \pi \eps}, \qquad
 x P_\eps(x) = \frac{\sin(\vartheta/2) \cos(\vartheta/2)}{\pi} = \frac{\sin \vartheta}{2 \pi} .
\]
By part (a), $P_\eps^{-1} \mathcal V$ is spanned by the functions $P_\eps^j$ with $j \ge 0$ and $x P_\eps^j$ with $j \ge 1$. Using $x P_\eps^j = (x P_\eps) P_\eps^{j - 1}$ and
\[
 (x P_\eps)^2 = \frac{\eps}{\pi} P_\eps - \eps^2 P_\eps^2 ,
\]
we see that this span is the unital algebra generated by $P_\eps$ and $x P_\eps$. By \eqref{e:proc-cayley}, it is the unital algebra generated by $\cos \vartheta$ and $\sin \vartheta$, which is the algebra of real trigonometric polynomials.

We now prove part (c). The numerator of \eqref{e:proc-ratio} satisfies
\[
 \prod_{j = 1}^k \bigl( (x - u_j)^2 + b_j^2 \bigr) = x^{2k} - 2 \Bigl( \sum_{j = 1}^k u_j \Bigr) x^{2k - 1} + O(x^{2k - 2}) = x^{2k} + O(x^{2k - 2}) ,
\]
where $O(x^{2k - 2})$ denotes a polynomial of degree at most $2k - 2$, and the even polynomial $(x^2 + \eps^2)^k$ is also monic of degree $2k$. Their difference has degree at most $2k - 2$, and
\[
 R_{k, \mathbf u, \mathbf b} - 1 = \Bigl( \frac \pi \eps \Bigr)^k \Bigl( \prod_{j = 1}^k \bigl( (x - u_j)^2 + b_j^2 \bigr) - (x^2 + \eps^2)^k \Bigr) P_\eps^k \in \mathcal V .
\]
(The balance condition $\sum_j u_j = 0$ is exactly what is needed here.)

Next, the function $R_{k, \mathbf u, \mathbf b}$ is continuous, positive since $b_j > 0$, and tends to one at $\pm \infty$. It is bounded below by some $c_R > 0$. Every element $a P_\eps^k$ of $\mathcal V$ satisfies $|a P_\eps^k| \le C P_\eps$, since $|a| P_\eps^{k - 1}$ is bounded when $\deg a \le 2k - 2$. Using $|\log y| \le |y - 1|/\min(y, 1)$ for $y > 0$, we conclude that
\[
 |\log R_{k, \mathbf u, \mathbf b}| \le c_R^{-1} |R_{k, \mathbf u, \mathbf b} - 1| \le C_R P_\eps . \qedhere
\]
\end{proof}

\section{Tightness and subsequential limits}\label{s:proc-tight}

We now restate the matrix inputs in the unfolded variables. By \eqref{e:iface-counts-unfolded}, there exists $C_{\exp} > 0$, depending only on $E_*$, such that for all sufficiently large $N$, all bounded intervals $I \subset \R$, and all $m \ge 1$,
\begin{equation}\label{e:proc-counts}
 \E \bigl( \Xi_N(I) \bigr)_m \le (C_{\exp} |I|)^m .
\end{equation}
By \eqref{e:iface-intensity}, the measures $\E \Xi_N$ converge vaguely to the Lebesgue measure. We say that a random point measure $\Xi$ satisfies the \emph{exponential bound with constant $C_{\exp}$} if
\begin{equation}\label{e:proc-exp}
 \E e^{t \Xi(I)} \le \exp \bigl\{ C_{\exp} |I| (e^t - 1) \bigr\} \qquad \text{for all } t \ge 0 \text{ and all bounded intervals } I \subset \R .
\end{equation}

\begin{proposition}\label{p:proc-subsequences}
The laws of the processes $\Xi_N$ are tight in $\mathcal N(\R)$, and $\Xi_N$ satisfies \eqref{e:proc-exp} for all sufficiently large $N$. Let $\Xi$ be the limit in law of $\Xi_N$ along a subsequence. Then $\Xi$ satisfies \eqref{e:proc-exp}, it is stationary and simple, and $\E \Xi(I) = |I|$ for all bounded intervals $I$. Moreover, $\Xi_N(I) \to \Xi(I)$ in law along the subsequence for all bounded intervals $I$.
\end{proposition}

\begin{proof}
We begin with the exponential bound for $\Xi_N$. For an integer $n \ge 0$ and $t \ge 0$, the binomial theorem gives
\[
 e^{t n} = \bigl( 1 + (e^t - 1) \bigr)^n = \sum_{m = 0}^n \frac{(n)_m}{m!} (e^t - 1)^m .
\]
Applying this with $n = \Xi_N(I)$ and using \eqref{e:proc-counts}, we find
\[
 \E e^{t \Xi_N(I)} = \sum_{m \ge 0} \frac{(e^t - 1)^m}{m!} \E \bigl( \Xi_N(I) \bigr)_m \le \sum_{m \ge 0} \frac{(C_{\exp} |I| (e^t - 1))^m}{m!} = \exp \bigl\{ C_{\exp} |I| (e^t - 1) \bigr\} .
\]
In particular,
\[
 \sup_N \E \Xi_N(I) \le C_{\exp} |I| \qquad \text{for all bounded intervals } I ,
\]
where the supremum is over all sufficiently large $N$. Markov's inequality then implies that the counts $\Xi_N(I)$ are tight, and a family of random point measures whose counts of bounded intervals are tight is tight in $\mathcal N(\R)$ \cite[Chapter~11]{daley2008introduction}.

Now let $\Xi$ be a subsequential limit. Recall that for a bounded open interval $I$, the map $\zeta \mapsto \zeta(I)$ is lower semicontinuous on $\mathcal N(\R)$. Since it takes integer values, the map $\zeta \mapsto F(\zeta(I))$ is lower semicontinuous for every nondecreasing $F \colon \{0, 1, 2, \dots\} \to [0, \infty)$. By the portmanteau theorem for nonnegative lower semicontinuous functions, we have
\begin{equation}\label{e:proc-fatou}
 \E F \bigl( \Xi(I) \bigr) \le \liminf_{N} \E F \bigl( \Xi_N(I) \bigr) ,
\end{equation}
where the limit is taken along the subsequence. With $F(n) = e^{t n}$, we obtain \eqref{e:proc-exp} for $\Xi$ and open intervals. Approximating a bounded interval from outside by open intervals, we extend this bound to all bounded intervals. With $F(n) = (n)_2$, which is nondecreasing on the nonnegative integers, we find
\begin{equation}\label{e:proc-pairs}
 \E \bigl( \Xi(I) \bigr)_2 \le (C_{\exp} |I|)^2 \qquad \text{for all bounded open intervals } I .
\end{equation}

Next, the case $F(n) = n$ of \eqref{e:proc-fatou} implies that, for all $a \in \R$ and $\eta > 0$,
\[
 \E \Xi(\{a\}) \le \E \Xi \bigl( (a - \eta, a + \eta) \bigr) \le 2 C_{\exp} \eta .
\]
Letting $\eta \downarrow 0$, we see that $\Xi(\{a\}) = 0$ almost surely for every deterministic $a$. The boundary of a bounded interval $I$ is thus almost surely not charged by $\Xi$, the map $\zeta \mapsto \zeta(I)$ is almost surely continuous at $\Xi$, and $\Xi_N(I) \to \Xi(I)$ in law along the subsequence. By \eqref{e:proc-counts} with $m = 2$, the variables $\Xi_N(I)$ are bounded in $L^2$, hence uniformly integrable, and \eqref{e:iface-intensity} implies that $\E \Xi_N(I) \to |I|$. We conclude that $\E \Xi(I) = |I|$.

We now prove stationarity. For $\zeta \in \mathcal N(\R)$ and $s \in \R$, we let $\zeta + s$ denote the image of $\zeta$ under the translation $x \mapsto x + s$, so that
\[
 (\zeta + s)(A) = \zeta(A - s)
\]
for Borel sets $A$. The map $\zeta \mapsto \zeta + s$ is continuous on $\mathcal N(\R)$. Adding $s a_N^{-1} \Id$ to $H$ shifts every eigenvalue by $s/a_N$, and every unfolded eigenvalue by $s$.

Let $\Phi$ be a bounded continuous function on $\mathcal N(\R)$. For a Hermitian matrix $M$ with eigenvalues $\lambda_\alpha(M)$, we set
\[
 F(M) = \Phi \Bigl( \sum_\alpha \delta_{a_N (\lambda_\alpha(M) - E)} \Bigr) .
\]
Then $F(H) = \Phi(\Xi_N)$ and $F(H + s a_N^{-1} \Id) = \Phi(\Xi_N + s)$. Applying \eqref{e:iface-shift} with this $F$ and with $b = s$, and using the last assertion of \zcref{l:iface-gaussian}, we find
\[
 \bigl| \E \Phi(\Xi_N + s) - \E \Phi(\Xi_N) \bigr| \le \sqrt{2/\pi} \, \|\Phi\|_\infty \frac{|s|}{a_N \sigma_N} \le C \|\Phi\|_\infty |s| W^{-1/2} .
\]
The function $\zeta \mapsto \Phi(\zeta + s)$ is bounded and continuous. Letting $N \to \infty$ along the subsequence, we find
\[
 \E \Phi(\Xi + s) = \E \Phi(\Xi) .
\]
The function $\Phi$ is arbitrary, and $\Xi + s$ and $\Xi$ have the same law for every $s \in \R$.

Finally, we prove that $\Xi$ is simple. Fix $A > 0$ and $\delta \in (0, 1)$, and for $l \in \Z$ let
\[
 I_l = \bigl( (l - 1) \delta, (l + 2) \delta \bigr) .
\]
Let $x \le y$ be points of $[-A, A]$ with $y - x \le \delta$, and let $l = \lfloor x/\delta \rfloor$. Then
\begin{equation}\label{e:proc-cover}
 (l - 1) \delta < l \delta \le x \le y \le x + \delta < (l + 2) \delta ,
\end{equation}
so that $x$ and $y$ both lie in $I_l$. Consequently, if $\Xi$ has a multiple point in $[-A, A]$, then $\Xi(I_l) \ge 2$ for some $l$ with $|l| \le A/\delta + 1$. Using $\1\{n \ge 2\} \le (n)_2/2$ for integers $n \ge 0$ and \eqref{e:proc-pairs} for the intervals $I_l$, of length $3 \delta$, we obtain
\[
 \Prob \bigl( \Xi \text{ has a multiple point in } [-A, A] \bigr) \le \sum_{|l| \le A/\delta + 1} \frac{\E (\Xi(I_l))_2}{2} \le \Bigl( \frac{2A}{\delta} + 3 \Bigr) \frac{(3 C_{\exp} \delta)^2}{2} ,
\]
which tends to zero as $\delta \downarrow 0$. The number $A$ is arbitrary, and $\Xi$ is simple.
\end{proof}

We will also use the following consequences of \eqref{e:proc-exp}.

\begin{lemma}\label{l:proc-moments}
Let $\Xi$ be a random point measure that satisfies \eqref{e:proc-exp} with constant $C_{\exp}$.
\begin{enumerate}[label=(\alph*)]
\item For every measurable $g \ge 0$ on $\R$ such that $A_g = \sum_{j \in \Z} \sup_{[j, j + 1)} g < \infty$,
\begin{equation}\label{e:proc-jensen}
 \E e^{\Xi(g)} \le \exp \bigl\{ C_{\exp} (e^{A_g} - 1) \bigr\} .
\end{equation}
\item Let $(\Xi^{(n)})_{n \ge 1}$ be random point measures that satisfy \eqref{e:proc-exp} with the constant $C_{\exp}$ and converge in law to $\Xi$. Let $g_1, \dots, g_l$ be continuous functions on $\R$ such that $|g_i| \le c P_\eps$ for some $c > 0$, and write $\Xi(\mathbf g) = (\Xi(g_1), \dots, \Xi(g_l))$. Then $\Xi^{(n)}(\mathbf g) \to \Xi(\mathbf g)$ in law. Moreover, $\E F(\Xi^{(n)}(\mathbf g)) \to \E F(\Xi(\mathbf g))$ for every continuous $F \colon \R^l \to \C$ such that $|F(y)| \le C_F e^{C_F |y|}$ for some $C_F > 0$.
\end{enumerate}
\end{lemma}

\begin{proof}
For part (a), let $I_j = [j, j + 1)$ and $w_j = \sup_{I_j} g$. The weights $w_j/A_g$ sum to one, and
\[
 \Xi(g) \le \sum_{j \in \Z} \frac{w_j}{A_g} \, A_g \Xi(I_j) .
\]
By the convexity of the exponential and \eqref{e:proc-exp} with $t = A_g$, we have
\[
 \E e^{\Xi(g)} \le \sum_{j \in \Z} \frac{w_j}{A_g} \E e^{A_g \Xi(I_j)} \le \exp \bigl\{ C_{\exp} (e^{A_g} - 1) \bigr\} .
\]

For part (b), we first note that Jensen's inequality and \eqref{e:proc-exp} imply that
\[
 e^{t \E \Xi(I)} \le \E e^{t \Xi(I)} \le \exp \bigl\{ C_{\exp} |I| (e^t - 1) \bigr\}, \qquad \text{so that} \qquad \E \Xi(I) \le C_{\exp} |I| \, \frac{e^t - 1}{t} \to C_{\exp} |I|
\]
as $t \downarrow 0$. For $A \ge 3$, let $\chi_A \in C_c(\R)$ satisfy $0 \le \chi_A \le 1$ and $\chi_A = 1$ on $[-A, A]$. Then
\begin{equation}\label{e:proc-tail}
 \E \bigl| \Xi^{(n)} \bigl( (1 - \chi_A) g_i \bigr) \bigr| \le c \sum_{|j| \ge A - 1} \E \Xi^{(n)}(I_j) \sup_{I_j} P_\eps \le \frac{C}{A}
\end{equation}
uniformly in $n$, where we used the previous bound for the intervals $I_j$, of length one. The same bound holds for $\Xi$.

Now let $F$ be bounded and Lipschitz. The map $\zeta \mapsto \zeta(\chi_A \mathbf g)$ is continuous on $\mathcal N(\R)$, since $\chi_A g_i \in C_c(\R)$. Using this continuity and \eqref{e:proc-tail}, we obtain
\[
\begin{aligned}
&\limsup_{n \to \infty} \bigl| \E F(\Xi^{(n)}(\mathbf g)) - \E F(\Xi(\mathbf g)) \bigr| \\
 &\qquad \le \limsup_{n \to \infty} \bigl| \E F(\Xi^{(n)}(\chi_A \mathbf g)) - \E F(\Xi(\chi_A \mathbf g)) \bigr| + \frac{2 C l \operatorname{Lip}(F)}{A}
 = \frac{2 C l \operatorname{Lip}(F)}{A} ,
\end{aligned}
\]
and letting $A \to \infty$ proves the convergence in law.

For the second assertion, part (a) with $g = t c l P_\eps$ yields, for all $t \ge 0$,
\[
 \sup_n \E e^{t |\Xi^{(n)}(\mathbf g)|} \le \sup_n \E e^{t c l \, \Xi^{(n)}(P_\eps)} \le \exp \bigl\{ C_{\exp} \bigl( e^{t c l A_{P_\eps}} - 1 \bigr) \bigr\} < \infty ,
\]
where $|y|$ denotes the $\ell^1$ norm of $y \in \R^l$ and $A_{P_\eps} = \sum_j \sup_{I_j} P_\eps$. Taking $t = 2 C_F$, we obtain
\[
 \sup_n \E \bigl| F(\Xi^{(n)}(\mathbf g)) \bigr|^2 \le C_F^2 \sup_n \E e^{2 C_F |\Xi^{(n)}(\mathbf g)|} < \infty .
\]
The variables $F(\Xi^{(n)}(\mathbf g))$ are therefore uniformly integrable, and their expectations converge, by the convergence in law.
\end{proof}

\section{A determining class}\label{s:proc-determination}

We now show that the products over the ratios \eqref{e:proc-ratio} determine the law of a point process. The main step is the following approximation lemma, in which the rank $k$ depends on $f$ and $\eta$.

\begin{lemma}\label{l:proc-rational}
Let $f \in C_c(\R)$ be nonnegative and let $\eta > 0$. Then there exist $k \ge 1$, $\mathbf u \in \R^k$ with $\sum_j u_j = 0$ and $\mathbf b \in (0, \infty)^k$ such that $R = R_{k, \mathbf u, \mathbf b}$ satisfies
\begin{equation}\label{e:proc-weighted-approx}
 c_f \le R \le 1 \quad \text{and} \quad |\log R(x) + f(x)| \le \eta P_\eps(x) \quad \text{for all } x \in \R ,
\end{equation}
where $c_f = e^{-\|f\|_\infty}/2$.
\end{lemma}

\begin{proof}
We define
\[
 \psi_f = \frac{1 - e^{-f}}{P_\eps} ,
\]
which is nonnegative, continuous, and compactly supported. As a function of the Cayley angle $\vartheta$ of \zcref{l:proc-algebra}(b), the function $\sqrt{\psi_f}$ vanishes near $\vartheta = \pm \pi$ and extends to a continuous function on the circle. By Fej\'er's theorem \cite[Chapter~I]{katznelson2004introduction}, the Ces\`aro means of the Fourier series of $\sqrt{\psi_f}$ are real trigonometric polynomials $q$ that converge to $\sqrt{\psi_f}$ uniformly. In particular, the number
\[
 \delta = \|q^2 - \psi_f\|_\infty \le \|q - \sqrt{\psi_f}\|_\infty \bigl( \|q\|_\infty + \|\sqrt{\psi_f}\|_\infty \bigr)
\]
can be made arbitrarily small. By \zcref{l:proc-algebra}(b), each such $q$ belongs to $P_\eps^{-1} \mathcal V$. Every element of $\mathcal V$ has the form $b P_\eps^{M + 1}$ with $M \ge 0$ and $\deg b \le 2M$, and therefore there exist $M \ge 0$ and $a \in \R[x]$ with $\deg a \le 2M$ such that $q = a P_\eps^M$. We set
\begin{equation}\label{e:proc-constructR}
 R = 1 - q^2 P_\eps = 1 - a^2 P_\eps^{2M + 1} .
\end{equation}

We first prove \eqref{e:proc-weighted-approx}. By the definition of $\psi_f$, we have
\[
 R - e^{-f} = P_\eps (\psi_f - q^2), \qquad \text{and therefore} \qquad |R - e^{-f}| \le \delta P_\eps .
\]
We have $R \le 1$, since $q^2 P_\eps \ge 0$. Using $P_\eps \le (\pi \eps)^{-1}$, we find
\[
 R \ge e^{-\|f\|_\infty} - \frac{\delta}{\pi \eps} \ge c_f \qquad \text{if } \delta \le \frac{\pi \eps}{2} e^{-\|f\|_\infty} .
\]
Since $R$ and $e^{-f}$ take values in $[c_f, 1]$, on which the logarithm has the Lipschitz constant $c_f^{-1}$, we obtain
\[
 |\log R + f| = |\log R - \log e^{-f}| \le \frac{|R - e^{-f}|}{c_f} \le \frac{\delta}{c_f} P_\eps ,
\]
which proves \eqref{e:proc-weighted-approx} once $\delta \le \eta c_f$.

It remains to show that $R$ has the form \eqref{e:proc-ratio}. Let $k = 2M + 1$. Using $P_\eps^k = (\eps/\pi)^k (x^2 + \eps^2)^{-k}$, we write $R = \mathsf p/(x^2 + \eps^2)^k$ with
\[
 \mathsf p(x) = (x^2 + \eps^2)^k - \Bigl( \frac \eps \pi \Bigr)^k a(x)^2 , \qquad \deg (a^2) \le 4M = 2k - 2 .
\]
The polynomial $\mathsf p$ is monic of degree $2k$, with vanishing coefficient of $x^{2k - 1}$. Further, $\mathsf p$ is positive on $\R$, since $R \ge c_f$. Having no real roots, it has $k$ pairs of complex conjugate roots $u_j \pm \iu b_j$ with $b_j > 0$, counted with multiplicity. This gives the factorization
\[
 \mathsf p(x) = \prod_{j = 1}^k \bigl( (x - u_j)^2 + b_j^2 \bigr), \qquad \sum_{j = 1}^k u_j = 0 ,
\]
where the second identity follows from the vanishing coefficient of $x^{2k - 1}$. This shows that $R = R_{k, \mathbf u, \mathbf b}$.
\end{proof}

\begin{proposition}\label{p:proc-determination}
The following statements hold.
\begin{enumerate}[label=(\roman*)]
\item Let $\Xi$ and $\Xi'$ be random point measures such that $\E \Xi(P_\eps) + \E \Xi'(P_\eps) < \infty$. Suppose that $\E \prod_{x \in \Xi} R(x) = \E \prod_{x \in \Xi'} R(x)$ for all ratios $R = R_{k, \mathbf u, \mathbf b}$ of the form \eqref{e:proc-ratio} with $R \le 1$. Then $\Xi$ and $\Xi'$ have the same law.
\item Let $\Xi$ and $(\Xi^{(n)})_{n \ge 1}$ be random point measures that satisfy \eqref{e:proc-exp} with a common constant $C_{\exp}$, and suppose that $\Xi^{(n)} \to \Xi$ in law. Then $\E \prod_{x \in \Xi^{(n)}} R(x) \to \E \prod_{x \in \Xi} R(x)$ for all ratios $R$ of the form \eqref{e:proc-ratio}, including those that exceed one somewhere.
\end{enumerate}
\end{proposition}

\begin{proof}
Recall from \zcref{l:proc-algebra}(c) that each ratio $R$ satisfies $|\log R| \le C_R P_\eps$. Whenever $\Xi(P_\eps) < \infty$, this bound shows that the product
\[
 \prod_{x \in \Xi} R(x) = \exp \Xi(\log R), \qquad |\Xi(\log R)| \le C_R \, \Xi(P_\eps) ,
\]
is well defined.

For part (i), let $f \in C_c(\R)$ be nonnegative, let $\eta > 0$, and choose $R$ by \zcref{l:proc-rational}. The functions $f$ and $-\log R$ are nonnegative, and $|e^{-s} - e^{-s'}| \le |s - s'|$ for $s, s' \ge 0$. Using \eqref{e:proc-weighted-approx}, we have
\[
 \Bigl| \E e^{-\Xi(f)} - \E \prod_{x \in \Xi} R(x) \Bigr| \le \E \Xi \bigl( |f + \log R| \bigr) \le \eta \, \E \Xi(P_\eps) ,
\]
and the same bound holds for $\Xi'$. Letting $\eta \to 0$, we find
\[
 \E e^{-\Xi(f)} = \E e^{-\Xi'(f)} .
\]
The Laplace functionals of the nonnegative functions in $C_c(\R)$ determine the law of a random measure \cite[Chapter~9]{daley2008introduction}, and therefore $\Xi$ and $\Xi'$ have the same law.

To prove part (ii), we apply \zcref{l:proc-moments}(b) with $l = 1$, $g_1 = \log R$, and $F(y) = e^y$; its hypotheses hold, since $\log R$ is continuous and $|\log R| \le C_R P_\eps$.
\end{proof}

\section{Continuation to all source parameters}\label{s:proc-continuation}

For $k \ge 1$, we parametrize the sources by
\begin{equation}\label{e:proc-parameters}
 \mathbf p = (u_1, \dots, u_{k - 1}, b_1, \dots, b_k) \in \C^{2k - 1}, \qquad u_k = -\sum_{j < k} u_j ,
\end{equation}
and for complex $\mathbf p$ we define $R_{k, \mathbf p}$ by the formula \eqref{e:proc-ratio}, as a function of real $x$. The real parameters $\mathbf p \in \R^{k - 1} \times (0, \infty)^k$ correspond to the sources of \eqref{e:proc-ratio}, and for them $R_{k, \mathbf p} = R_{k, \mathbf u, \mathbf b}$. We write $T_N^{(k)}(\mathbf p) = \E \prod_{x \in \Xi_N} R_{k, \mathbf p}(x)$, and $\mathcal Z^{(k)}_{X, \eps}(\mathbf p) = \mathcal Z^{(k)}_{X, \eps}(\mathbf u, \mathbf b)$ for $\mathbf p \in \R^{k - 1} \times (0, \infty)^k$.

We first deduce from \zcref{l:sigma-entire} that the partition function is entire in these parameters.

\begin{lemma}\label{l:proc-entire}
For every $k \ge 1$, the function $\mathbf p \mapsto \mathcal Z^{(k)}_{X, \eps}(\mathbf p)$ on $\R^{k - 1} \times (0, \infty)^k$ is the restriction of an entire function on $\C^{2k - 1}$, which we also denote by $\mathcal Z^{(k)}_{X, \eps}$.
\end{lemma}

\begin{proof}
For $\mathbf p \in \C^{2k - 1}$, with $u_k$ as in \eqref{e:proc-parameters}, we let $\Gamma(\mathbf p)$ denote the source supermatrix of \zcref{d:rep-sources} with the fermion-fermion block
\[
 \Gamma_{\mathrm F}(\mathbf p) = \diag \bigl( b_1 - \iu u_1, \dots, b_k - \iu u_k, -b_1 - \iu u_1, \dots, -b_k - \iu u_k \bigr) .
\]
The diagonal entries of $\Gamma_{\mathrm F}(\mathbf p)$ are affine functions of $\mathbf p$, and for $\mathbf p \in \R^{k - 1} \times (0, \infty)^k$ we have $\Gamma(\mathbf p) = \Gamma(\mathbf u, \mathbf b)$. We apply \zcref{l:sigma-entire}\ref{i:sigma-entire-series} with the present $k$, $X$, and $\eps$. It shows that the partition function, as a function of the diagonal entries of $\Gamma_{\mathrm F}$, is the restriction of an entire function on $\C^{2k}$. The composition of this entire function with the affine map $\mathbf p \mapsto \Gamma_{\mathrm F}(\mathbf p)$ is entire on $\C^{2k - 1}$, and it agrees with $\mathcal Z^{(k)}_{X, \eps}(\mathbf p)$ for $\mathbf p \in \R^{k - 1} \times (0, \infty)^k$.
\end{proof}

The next lemma is a version of Vitali's theorem in several variables, for the expectations of products over point processes with uniform exponential bounds.

\begin{lemma}\label{l:proc-continuation}
Let $(\Xi^{(n)})_{n \ge 1}$ be random point measures that satisfy \eqref{e:proc-exp} with a common constant $C_{\exp}$. Fix $k \ge 1$, and set $T_n(\mathbf p) = \E \prod_{x \in \Xi^{(n)}} R_{k, \mathbf p}(x)$ for $\mathbf p \in \C^{2k - 1}$. Then every $T_n$ is entire, and
\[
 \sup_n \sup_{\mathbf p \in K} |T_n(\mathbf p)| < \infty \qquad \text{for every compact } K \subset \C^{2k - 1} .
\]
If $T_n$ converges pointwise on a nonempty open subset of $\R^{2k - 1}$, then $T_n$ converges locally uniformly on $\C^{2k - 1}$, together with all its partial derivatives, to an entire function.
\end{lemma}

\begin{proof}
The numerator of $R_{k, \mathbf p}$ is a polynomial in $(x, \mathbf p)$. It is monic of degree $2k$ in $x$, with vanishing coefficient of $x^{2k - 1}$, since $\sum_j u_j = 0$. As in the proof of \zcref{l:proc-algebra}(c), its difference with $(x^2 + \eps^2)^k$ has degree at most $2k - 2$ in $x$, with coefficients that are polynomials in $\mathbf p$. It follows that, for every compact $K \subset \C^{2k - 1}$, there exists $C_K$ such that
\begin{equation}\label{e:proc-complexbound}
 |R_{k, \mathbf p}(x) - 1| \le C_K P_\eps(x), \qquad x \in \R, \ \mathbf p \in K .
\end{equation}

Let $\zeta \in \mathcal N(\R)$ satisfy $\zeta(P_\eps) < \infty$. By \eqref{e:proc-complexbound}, the product $\prod_{x \in \zeta} R_{k, \mathbf p}(x)$ converges absolutely and uniformly for $\mathbf p \in K$. It is entire in $\mathbf p$, as a locally uniform limit of polynomials, and
\[
 \sup_{\mathbf p \in K} \Bigl| \prod_{x \in \zeta} R_{k, \mathbf p}(x) \Bigr| \le \exp \bigl\{ C_K \zeta(P_\eps) \bigr\} .
\]
Using \eqref{e:proc-jensen} with $g = C_K P_\eps$, we find that the expectation of the right side is bounded uniformly in $n$,
\[
 \sup_n \E \exp \bigl\{ C_K \Xi^{(n)}(P_\eps) \bigr\} \le \exp \bigl\{ C_{\exp} \bigl( e^{C_K A_{P_\eps}} - 1 \bigr) \bigr\} ,
\]
where $A_{P_\eps}$ is as in the proof of \zcref{l:proc-moments}. Dominated convergence shows that $T_n$ is continuous, and Fubini's theorem and Morera's theorem in each variable show that $T_n$ is holomorphic in each variable separately. By Osgood's lemma, $T_n$ is entire, and the uniform bound on compact sets holds.

Next, Cauchy's estimates show that the functions $T_n$ are equicontinuous on compact sets. The Arzel\`a--Ascoli theorem then shows that every subsequence of $(T_n)$ has a further subsequence that converges locally uniformly on $\C^{2k - 1}$, and the limit is entire (this is Montel's theorem in several variables). Let $T$ and $T'$ be two such limits, which agree on a nonempty open subset $U$ of $\R^{2k - 1}$. The complex partial derivatives of $T - T'$ of every order at a point of $U$ are equal to its real partial derivatives in the directions of $\R^{2k - 1}$, which vanish. We have
\[
 \partial^\beta (T - T')(\mathbf p) = 0 \qquad \text{for all multi-indices } \beta \text{ and } \mathbf p \in U ,
\]
the Taylor series of $T - T'$ at a point of $U$ vanishes, and since $\C^{2k - 1}$ is connected, $T = T'$. We conclude that the whole sequence $(T_n)$ converges locally uniformly. Cauchy's estimates then imply the locally uniform convergence of all partial derivatives.
\end{proof}

\begin{corollary}\label{c:proc-complex-source}
For every $k \ge 1$, $T_N^{(k)} \to \mathcal Z^{(k)}_{X, \eps}$ locally uniformly on $\C^{2k - 1}$, together with all partial derivatives. In particular, $T_N^{(k)}(\mathbf u, \mathbf b) \to \mathcal Z^{(k)}_{X, \eps}(\mathbf u, \mathbf b)$ for all $\mathbf u \in \R^k$ with $\sum_j u_j = 0$ and all $\mathbf b \in (0, \infty)^k$.
\end{corollary}

\begin{proof}
\zcref[S]{p:proc-subsequences} shows that the processes $\Xi_N$ satisfy \eqref{e:proc-exp} with a constant independent of $N$, for all sufficiently large $N$. Let
\[
 U = \bigl\{ \mathbf p \in \R^{k - 1} \times (0, \infty)^k : \|\mathbf u\| + \|\mathbf b - \eps \1\| < r_{k, \eps} \bigr\} ,
\]
which is an open subset of $\R^{2k - 1}$ containing $(0, \eps \1)$; on $U$, the partition function $\mathcal Z^{(k)}_{X, \eps}(\mathbf p)$ is defined by \zcref{d:sigma-partition}. \zcref[S]{t:interface}(a) shows that $T_N^{(k)}$ converges to $\mathcal Z^{(k)}_{X, \eps}$ on $U$. We apply \zcref{l:proc-continuation} to the processes $\Xi_N$ in place of $\Xi^{(n)}$, and we obtain the locally uniform convergence of $T_N^{(k)}$ on $\C^{2k - 1}$, with all partial derivatives, to an entire function $T$. By \zcref{l:proc-entire}, the function $\mathcal Z^{(k)}_{X, \eps}$ is entire. It agrees with $T$ on $U$, and the argument in the proof of \zcref{l:proc-continuation} shows that $T = \mathcal Z^{(k)}_{X, \eps}$ everywhere.
\end{proof}

\section{Proof of parts (a) and (b)}\label{s:proc-proof}

We now prove the statements of \zcref{t:process} about $\Xi_N$ and its source law.

\begin{proof}[Proof of \zcref{t:process}(a) and (b)]
By \zcref{p:proc-subsequences}, the laws of $\Xi_N$ are tight. Let $\Xi$ be a subsequential limit. The same proposition shows that $\Xi$ is stationary, simple, and of unit intensity, and $\Xi$ and $\Xi_N$ satisfy \eqref{e:proc-exp} with a common constant.

Fix $k \ge 1$, $\mathbf u \in \R^k$ with $\sum_j u_j = 0$, and $\mathbf b \in (0, \infty)^k$. By \zcref{c:proc-complex-source}, we have $T_N^{(k)}(\mathbf u, \mathbf b) \to \mathcal Z^{(k)}_{X, \eps}(\mathbf u, \mathbf b)$, which is the first assertion of part (b). Next, we apply \zcref{p:proc-determination}(ii) along the subsequence, with the processes $\Xi_N$ in place of $\Xi^{(n)}$, and we obtain
\begin{equation}\label{e:proc-source-law}
 \E \prod_{x \in \Xi} \prod_{j = 1}^k \frac{(x - u_j)^2 + b_j^2}{x^2 + \eps^2} = \mathcal Z^{(k)}_{X, \eps}(\mathbf u, \mathbf b) .
\end{equation}
Since $\Xi$ has unit intensity, we have
\[
 \E \Xi(P_\eps) = \int_\R P_\eps(x) \dd x = 1 ,
\]
and \zcref{p:proc-determination}(i) implies that the values \eqref{e:proc-source-law}, for all $k$, $\mathbf u$, and $\mathbf b$, determine the law of $\Xi$. These values are functions of $X$ and $\eps$ alone, so all subsequential limits have the same law. We let $\Xi_X$ denote a random point measure with this law.

The laws of $\Xi_N$ are tight and all their subsequential limits agree, and we conclude that $\Xi_N \to \Xi_X$ in law. Further, the law of $\Xi_X$ is the same for all sequences $(N, W, E)$ that tend to the critical limit at $X$, and \eqref{e:proc-source-law} at any one $\eps > 0$ determines it. This proves part (b) and all statements of part (a) except reflection invariance.

It remains to prove that $\Xi_X$ is invariant in law under the reflection $x \mapsto -x$. We may compute the law of $\Xi_X$ along a sequence with $E = 0$, for instance with $W = (N/(4X))^{1/2}$, so that $X_N(0) = X$. The law of $H$ is that of a centered Gaussian matrix, and $-H$ has the same law as $H$. Since $E = 0$, replacing $H$ by $-H$ replaces each $x_\alpha$ by $-x_\alpha$. Therefore $\Xi_N$ has the same law as its image under $x \mapsto -x$, and since this reflection is continuous on $\mathcal N(\R)$, the reflection invariance of $\Xi_X$ follows by passing to the limit.
\end{proof}

\section{The rank-one derivatives}\label{s:proc-jets}

We now prove part (c) of \zcref{t:process}. Throughout this section, $k = 1$, and we abbreviate
\[
 T_N(t) = T_N^{(1)}(0, \eps + t), \qquad \mathcal Z(t) = \mathcal Z^{(1)}_{X, \eps}(0, \eps + t), \qquad t \in \C .
\]
The only fermionic source parameter at rank one is $b_1 = \eps + t$, and \zcref{c:proc-complex-source} shows that $T_N \to \mathcal Z$ locally uniformly on $\C$, together with all derivatives. Part (c) is therefore equivalent to
\begin{equation}\label{e:proc-jet-values}
 \mathcal Z'(0) = 2 \pi, \qquad \mathcal Z''(0) = 4 \pi^2 \bigl( 1 + I_X(2 \pi \eps) \bigr) ,
\end{equation}
where we recall that $I_X$ is the function defined in \eqref{e:intro-I}. We prove the first identity on the side of the matrices, where it expresses the unit intensity of $\Xi_X$, and we obtain the second from the computation of rank one in the sigma model in \zcref{l:sigma-jets}.

\begin{proof}[Proof of the first identity in \eqref{e:proc-jet-values}]
For real $x$, we have
\[
 \partial_t \frac{x^2 + (\eps + t)^2}{x^2 + \eps^2} \Big|_{t = 0} = \frac{2 \eps}{x^2 + \eps^2} = 2 \pi P_\eps(x) .
\]
We differentiate the finite product in \eqref{e:intro-ratio} under the expectation, which is justified by the bound \eqref{e:proc-complexbound} and \eqref{e:proc-jensen}, as in the proof of \zcref{l:proc-continuation}. This gives
\[
 T_N'(0) = 2 \pi \E \sum_\alpha P_\eps(x_\alpha) = 2 \pi \E \Xi_N(P_\eps) .
\]

We now show that $\E \Xi_N(P_\eps) \to 1$. By parts (a) and (b), which we proved in \zcref{s:proc-proof}, we have $\Xi_N \to \Xi_X$ in law. By \zcref{p:proc-subsequences}, the processes $\Xi_N$, for all sufficiently large $N$, and $\Xi_X$ satisfy \eqref{e:proc-exp} with a common constant $C_{\exp}$. We apply \zcref{l:proc-moments}(b) with $l = 1$, $g_1 = P_\eps$, $c = 1$, and $F(y) = y$, and we obtain
\[
 \lim_{N \to \infty} \E \Xi_N(P_\eps) = \E \Xi_X(P_\eps) = \int_\R P_\eps(x) \dd x = 1 ,
\]
where the second equality holds since $\Xi_X$ has unit intensity, as in the proof of parts (a) and (b). Finally, \zcref{c:proc-complex-source} shows that $T_N' \to \mathcal Z'$, and we conclude that
\[
 \mathcal Z'(0) = \lim_{N \to \infty} T_N'(0) = 2 \pi . \qedhere
\]
\end{proof}

\begin{proof}[Proof of the second identity in \eqref{e:proc-jet-values}]
We apply \zcref{l:sigma-jets} with the present $X$ and $\eps$. The function $t \mapsto \mathcal Z^{(1)}_{X, \eps}(0, \eps + t)$ on $(-\eps, \infty)$ of that lemma is the restriction of the entire function $\mathcal Z$ of this section, by \zcref{l:proc-entire}, and the two have the same derivatives at $t = 0$. The second identity in \eqref{e:sigma-jets-values} is therefore the second identity in \eqref{e:proc-jet-values}. This completes the proof of \zcref{t:process}.
\end{proof}

The first identity in \eqref{e:sigma-jets-values} is the first identity in \eqref{e:proc-jet-values}, obtained within the sigma model from the one-insertion pairing in \eqref{e:sigma-jets-pairings}. The proof above identifies it, on the side of the matrices, with the unit intensity of $\Xi_X$.

\begin{remark}\label{r:proc-pair}
The second derivative has the following meaning for $\Xi_X$. For real $x$, we have
\[
 \partial_t^2 \frac{x^2 + (\eps + t)^2}{x^2 + \eps^2} \Big|_{t = 0} = \frac{2}{x^2 + \eps^2} = \frac{2\pi}{\eps} P_\eps(x) ,
\]
and differentiating the product in \eqref{e:intro-ratio} twice, we obtain
\[
 T_N''(0) = \frac{2 \pi}{\eps} \E \Xi_N(P_\eps) + 4 \pi^2 \E \bigl[ \Xi_N(P_\eps)^2 - \Xi_N(P_\eps^2) \bigr] .
\]
We apply \zcref{l:proc-moments}(b) with $g_1 = P_\eps$, $g_2 = P_\eps^2$, and $F(y_1, y_2) = y_1^2 - y_2$. It shows that the last expectation converges to $\E \sum_{x \ne y} P_\eps(x) P_\eps(y)$, where the sum runs over ordered pairs of distinct points of the simple process $\Xi_X$. Using \eqref{e:proc-jet-values} and the limit $\E \Xi_N(P_\eps) \to 1$ from its proof, we conclude that
\[
 \E \sum_{\substack{x, y \in \Xi_X \\ x \ne y}} P_\eps(x) P_\eps(y) = 1 + I_X(2 \pi \eps) - \frac{1}{2 \pi \eps} .
\]
This is the case $v = 0$ of \eqref{e:intro-pair-transform}, with $2 \eps$ in place of $\eps$, since $P_\eps * P_\eps = P_{2\eps}$ and $P_{2\eps}(0) = 1/(2\pi\eps)$; the general case is proved in \zcref{s:scales}.
\end{remark}