# Quartic SYK spectral edge — H3

Fix \(\beta>0\), \(M\ge1\), \(x_\alpha\ge0\), and \(v_\alpha\in\mathbb R\), with \(\sum_{\alpha=1}^M v_\alpha^2=\beta^2/4\).
For each even \(N\ge4\), let \(\chi_i,\eta_{i\alpha},\zeta_{i\alpha}\), \(1\le i\le N\), \(1\le\alpha\le M\), be self-adjoint Majorana operators in an irreducible representation: each squares to \(I/2\), and distinct operators anticommute. Put \(q_i=\sqrt2\chi_i\) and \(\Psi_S=-q_{i_1}q_{i_2}q_{i_3}q_{i_4}\) for \(S=\{i_1<i_2<i_3<i_4\}\).
Let \(J_S\) be independent standard real Gaussians, put \(c_N^2=3/(8N^3)\) with \(c_N>0\), and define
\[
A_{N,M,\theta}=\sqrt\theta\,\beta c_N\sum_{|S|=4}J_S\Psi_S
-\mathrm i\sum_{i,\alpha}\left(x_\alpha\eta_{i\alpha}\zeta_{i\alpha}
+\sqrt{1-\theta}\,v_\alpha\chi_i\eta_{i\alpha}\right),\qquad0\le\theta\le1.
\]
The trace \(\operatorname{tr}\) is normalized; expectation is over the Gaussians.

For a probability measure \(\mu\) on \([0,\infty)\), write
\[
G_\mu(u)=\int_0^\infty\frac{\cosh(x(1/2-u))}{2\cosh(x/2)}\,\mu(dx),\qquad0\le u\le1.
\]

Let \(B_{N,M,\theta}\) be obtained from \(A_{N,M,\theta}\) by removing every random quartic term incident to site \(N\), while retaining all bath terms.
Let \(r\ge1\), let \(\mathfrak p\) be a pairing of \(\{1,\ldots,2r\}\), and assign a distinct bulk label \(i\in\{1,\ldots,N-1\}\) to each pair. Write \(j_a\) for the label of the pair containing position \(a\).
For \(0<t_1<\cdots<t_{2r}<1\), define
\[
W_N(\mathbf t,\mathfrak p,\mathbf i)=
\frac{\mathbb E\operatorname{tr}\left(
e^{t_1B}q_{j_1}e^{(t_2-t_1)B}q_{j_2}\cdots
e^{(t_{2r}-t_{2r-1})B}q_{j_{2r}}e^{(1-t_{2r})B}\right)}
{\mathbb E\operatorname{tr}e^B},\qquad B=B_{N,M,\theta}.
\]
Put \(\varepsilon(\mathfrak p)=(-1)^{\operatorname{cr}(\mathfrak p)}\), where \(\operatorname{cr}\) counts pairs of arcs with endpoints \(a<c<b<d\).
For every sequence of even \(N_j\to\infty\) and \(\theta_j\to\theta_\infty\in[0,1]\), there exist a subsequence and one Borel probability measure \(\Pi\) on \(\mathcal P([0,\infty))\), with its weak topology, such that simultaneously for all fixed \(r\) and \(\mathfrak p\),
\[
W_{N_j}(\mathbf t,\mathfrak p,\mathbf i)\longrightarrow
\varepsilon(\mathfrak p)\int_{\mathcal P([0,\infty))}
\prod_{\{a,b\}\in\mathfrak p,\ a<b}2G_\mu(t_b-t_a)\,\Pi(d\mu).
\]
Convergence is uniform over the distinct bulk labels and locally uniform in the open simplex \(0<t_1<\cdots<t_{2r}<1\). The same \(\Pi\) applies to every \(r\), pairing, and parameter choice.