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\begin{document}

\title{
  Arrangement of nearby minima and saddles in the mixed $p$-spin energy landscape
}

\author{Jaron Kent-Dobias}
\affil{\textsc{DynSysMath}, Istituto Nazionale di Fisica Nucleare, Sezione di Roma}

\maketitle
\begin{abstract}
\end{abstract}

\cite{Ros_2020_Distribution, Ros_2019_Complex, Ros_2019_Complexity}

The mixed $p$-spin models are defined by the Hamiltonian
\begin{equation} \label{eq:hamiltonian}
  H(\mathbf s)=-\sum_p\frac1{p!}\sum_{i_1\cdots i_p}^NJ^{(p)}_{i_1\cdots i_p}s_{i_1}\cdots s_{i_p}
\end{equation}
where the vectors $\mathbf s\in\mathbb R^N$ are confined to the sphere
$\|\mathbf s\|^2=N$.  The coupling coefficients $J$ are fully-connected and random, with
zero mean and variance $\overline{(J^{(p)})^2}=a_pp!/2N^{p-1}$ scaled so that
the energy is typically extensive. The overbar denotes an average
over the coefficients $J$. The factors $a_p$ in the variances are freely chosen
constants that define the particular model. For instance, the `pure'
$p$-spin model has $a_{p'}=\delta_{p'p}$. This class of models encompasses all
statistically isotropic gaussian random Hamiltonians defined on the
hypersphere.

The covariance between the energy at two different points is a function of the overlap, or dot product, between those points, or
\begin{equation} \label{eq:covariance}
  \overline{H(\mathbf s_1)H(\mathbf s_2)}=Nf\left(\frac{\mathbf s_1\cdot\mathbf s_2}N\right)
\end{equation}
where the function $f$ is defined from the coefficients $a_p$ by
\begin{equation}
  f(q)=\frac12\sum_pa_pq^p
\end{equation}
In this paper, we will focus on models with a replica symmetric complexity.

We introduce the Kac--Rice \cite{Kac_1943_On, Rice_1944_Mathematical} measure
\begin{equation}
  d\nu_H(\mathbf s)
  =d\mathbf s\,\delta\big(\nabla H(\mathbf s)\big)\,
  \big|\det\operatorname{Hess}H(\mathbf s)\big|
\end{equation}
which counts stationary points of the function $H$. More interesting is the
measure conditioned on the energy density $E$ and stability $\mu$,
\begin{equation}
  d\nu_H(\mathbf s\mid E,\mu)
  =d\nu_H(\mathbf s)\,
  \delta\big(H(\mathbf s)-NE\big)\,
  \delta\big(N\mu-\operatorname{Tr}\operatorname{Hess}H(\mathbf s)\big)
\end{equation}


We want the typical number of stationary points with energy density
$E_2$ and stability $\mu_2$ that lie a fixed overlap $q$ from a reference
stationary point of energy density $E_1$ and stability $\mu_1$.
\begin{align*}
  \Sigma_{12}
  &=\frac1N\overline{\int\frac{d\nu_H(\mathbf s_0\mid E_0,\mu_0)}{\int d\nu_H(\mathbf s_0'\mid E_0,\mu_0)}\,
  \log\bigg(\int d\nu_H(\mathbf s_1\mid E_1,\mu_1)\,\delta(Nq-\mathbf s_0\cdot\mathbf s_1)\bigg)}
\end{align*}
\begin{align*}
  \Sigma_{12}
  &=\frac1N\lim_{n\to0}\lim_{m\to-1}\overline{\int d\nu_H(\mathbf s_0\mid E_0,\mu_0)\left(\int d\nu_H(\mathbf s_0'\mid E_0,\mu_0)\right)^m\,
  \frac\partial{\partial n}\bigg(\int d\nu_H(\mathbf s_1\mid E_1,\mu_1)\,\delta(Nq-\mathbf s_0\cdot \mathbf s_1)\bigg)^n}\\
  &=\frac1N\lim_{n\to0}\lim_{m\to0}\frac\partial{\partial n}\overline{\int\left(\prod_{b=1}^md\nu_H(\pmb\sigma_b\mid E_0,\mu_0)\right)\left(\prod_{a=1}^nd\nu_H(\mathbf s_a\mid E_1,\mu_1)\,\delta(Nq-\pmb \sigma_1\cdot \mathbf s_a)\right)}
\end{align*}
\begin{equation}
  \overline{\big|\det\operatorname{Hess}H(s)\big|\,\delta\big(N\mu-\operatorname{Tr}\operatorname{Hess}H(s)\big)}
  =e^{N\int d\lambda\,\rho(\lambda+\mu)\log|\lambda|}\delta(N\mu-s\cdot\partial H)
\end{equation}
\begin{equation}
  \rho(\lambda)=\begin{cases}
    \frac2{\pi}\sqrt{1-\big(\frac{\lambda}{\mu_\text m}\big)^2} & \lambda^2\leq\mu_\text m^2 \\
    0 & \text{otherwise}
  \end{cases}
\end{equation}
\begin{equation}
  \begin{aligned}
    \mathcal D(\mu)
    &=\int d\lambda\,\rho(\lambda+\mu)\ln|\lambda| \\
    &=\begin{cases}
      \frac12+\log\left(\frac12\mu_\text m\right)+\frac\mu{\mu_\text m}\left(\frac\mu{\mu_\text m}-\sqrt{\big(\frac\mu{\mu_\text m}\big)^2-1}\right)
      -\log\left(\frac{\mu}{\mu_\text m}-\sqrt{\big(\frac\mu{\mu_\text m}\big)^2-1}\right) & \mu>\mu_\text m \\
      \frac12+\log\left(\frac12\mu_\text m\right)+\frac{\mu^2}{\mu_\text m^2}
       & -\mu_\text m\leq\mu\leq\mu_\text m \\
       \frac12+\log\left(\frac12\mu_\text m\right)+\frac\mu{\mu_\text m}\left(\frac\mu{\mu_\text m}+\sqrt{\big(\frac\mu{\mu_\text m}\big)^2-1}\right)
      -\log\left(\frac{\mu}{\mu_\text m}+\sqrt{\big(\frac\mu{\mu_\text m}\big)^2-1}\right) & \mu<-\mu_\text m
    \end{cases}
  \end{aligned}
\end{equation}

\begin{align*}
  &\Sigma_{12}
  =\frac1N\lim_{n\to0}\lim_{m\to0}\frac\partial{\partial n}\int e^{Nm\mathcal S_0(\hat\beta_0,C^{00},R^{00},D^{00})+Nn\mathcal S_1(\hat\beta_0,\hat\beta_1,C^{00},C^{01},C^{11},R^{00},R^{01},R^{10},R^{11},D^{00},D^{01},D^{11})}
\end{align*}

\begin{align}
  C^{01}
  =\begin{bmatrix}
    q&\cdots&q\\
    q'&\cdots&q'\\
    \vdots&\ddots&\vdots\\
    q'&\cdots&q'
  \end{bmatrix}
  &&
  R^{01}
  =\begin{bmatrix}
    r_{01}&\cdots&r_{01}\\
    r_{01}'&\cdots&r_{01}'\\
    \vdots&\ddots&\vdots\\
    r_{01}'&\cdots&r_{01}'
  \end{bmatrix}
\end{align}

\begin{align*}
  &\Sigma_{12}
  =\frac1N\frac{e^{-\hat\beta_0E_0-r_0\mu_0+\frac12\left[\hat\beta_0^2f(1)-(2\hat\beta_0r_0^2+d_0)f'(1)+r_0^2f''(1)\right]+\mathcal D(\mu_0)}}{e^{N\Sigma(E_0,\mu_0)}}+\mathcal D(\mu_1)+\hat\beta_1E_1-\frac12\hat\mu_1-\mu_0r_{00}
  +\hat\beta_0\hat\beta_1f(q)+(\hat\beta_0r_{01}+\hat\beta_1r_{10}-d_{01})f'(q)+r_{01}r_{10}f''(q)
  \\&
  +\lim_{n\to0}\frac1n\bigg\{
    \frac12\sum_{ab}\left[
      \hat\beta_1^2f(C_{ab})+(2\hat\beta_1R_{ab}-D_{ab})f'(C_{ab})+R_{ab}^2f''(C_{ab})
    \right]
  \\&
    +\frac12\hat\mu_1\operatorname{Tr}C-\mu_1\operatorname{Tr}R
    +\frac12\log\det\begin{bmatrix}
      1&ir_{0}&q&ir_{01}\\
      ir_{0}&d_{0}&ir_{10}&d_{01}\\
      q&ir_{10}&C&iR\\
      ir_{01}&d_{01}&iR&D
    \end{bmatrix}
  \bigg\}
\end{align*}

\begin{align}
  \hat\beta_0
  &=-\frac{(\epsilon_0+\mu_0)f'(1)+\epsilon_0f''(1)}{f(1)\big(f'(1)+f''(1)\big)-f'(1)^2}\\
  r_0
  &=\frac{\mu_0f(1)+\epsilon_0f'(1)}{f(1)\big(f'(1)+f''(1)\big)-f'(1)^2} \\
  d_0
  &=\frac1{f'(1)}
  -\left(
    \frac{\mu_0f(1)+\epsilon_0f'(1)}{f(1)\big(f'(1)+f''(1)\big)-f'(1)^2}
    \right)^2
\end{align}

\begin{align*}
  \det\begin{bmatrix}
    1&ir_{0}&q&ir_{01}\\
    ir_{0}&d_{0}&ir_{10}&d_{01}\\
    q&ir_{10}&C&iR\\
    ir_{01}&d_{01}&iR&D
  \end{bmatrix}
  &=
  \det\begin{bmatrix}
    1&ir_{0}\\
    ir_{0}&d_{0}
  \end{bmatrix}
  \det\left(
    \begin{bmatrix}
      C&iR\\
      iR&D
    \end{bmatrix}
    -
    \begin{bmatrix}
      q&ir_{10}\\ir_{01}&d_{01}
    \end{bmatrix}
    \begin{bmatrix}
      1&ir_{0}\\
      ir_{0}&d_{0}
    \end{bmatrix}^{-1}
    \begin{bmatrix}
      q&ir_{01}\\ir_{10}&d_{01}
    \end{bmatrix}
  \right)
\end{align*}

\begin{align*}
  &
  \begin{bmatrix}
    \tilde c&\tilde r\\\tilde r&\tilde d
  \end{bmatrix}
  =
  \begin{bmatrix}
    q&ir_{10}\\ir_{01}&d_{01}
  \end{bmatrix}
  \begin{bmatrix}
    1&ir_{0}\\
    ir_{0}&d_{0}
  \end{bmatrix}^{-1}
  \begin{bmatrix}
    q&ir_{01}\\ir_{10}&d_{01}
  \end{bmatrix}\\
  &=
  \frac1{r_{0}^2+d_{0}}\begin{bmatrix}
    q^2d_{0}+2qr_{0}r_{10}-r_{10}^2
    &
    i\left[d_{01}(r_{10}-r_0q)+r_{01}(r_0r_{10}+d_0q)\right]
    \\
    i\left[d_{01}(r_{10}-r_0q)+r_{01}(r_0r_{10}+d_0q)\right]
    &
    d_{01}^2+2r_{0}r_{01}d_{01}-d_{0}r_{01}^2
  \end{bmatrix}
\end{align*}
This matrix with modify the diagonal of the RS matrix for the second spin.

Define $\tilde C=C-\tilde c$, $\tilde R=R-\tilde r$, $\tilde D=D-\tilde d$. Then
\begin{align*}
  &\Sigma_{12}
  =\mathcal D(\mu_1)+\hat\beta_1E_1-\hat\mu_1
  +\hat\beta_0\hat\beta_1f(q)+(\hat\beta_0r_{01}+\hat\beta_1r_{10}-d_{01})f'(q)+r_{01}r_{10}f''(q)
  \\&
  +\lim_{n\to0}\frac1n\bigg\{
    \frac12\sum_{ab}\left[
      \hat\beta_1^2f(C_{ab})+(2\hat\beta_1R_{ab}-D_{ab})f'(C_{ab})+R_{ab}^2f''(C_{ab})
    \right]
  \\&
    +\hat\mu_1\operatorname{Tr}C-\mu_1\operatorname{Tr}R
    +\frac12\log\det((C-\tilde c)(D-\tilde d)+(R-\tilde r)^2)
  \bigg\}
\end{align*}
These equations for $D^*$ are the same as those for the unpinned case, or
\[
  0=-\frac12f'(C)+\frac12((C-\tilde c)(D-\tilde d)+(R-\tilde r)^2)^{-1}(C-\tilde c)
\]
Solving, we get
\[
  D=\tilde d+f'(C)^{-1}-(C-\tilde c)^{-1}(R-\tilde r)^2
\]
\begin{align*}
  &\Sigma_{12}
  =\mathcal D(\mu_1)+\hat\beta_1E_1-\frac12\hat\mu_1
  +\hat\beta_0\hat\beta_1f(q)+(\hat\beta_0r_{01}+\hat\beta_1r_{10}-d_{01})f'(q)+r_{01}r_{10}f''(q)
  \\&
  +\lim_{n\to0}\frac1n\bigg\{
    \frac12\sum_{ab}\left[
      \hat\beta_1^2f(C_{ab})+(2\hat\beta_1R_{ab}-(f'(C)^{-1}_{ab}-((C-\tilde c)^{-1}(R-\tilde r)^2)_{ab}-\tilde d))f'(C_{ab})+R_{ab}^2f''(C_{ab})
    \right]
  \\&
    +\frac12\hat\mu_1\operatorname{Tr}C-\mu_1\operatorname{Tr}R
    +\frac12\log\det(C-\tilde c)-\frac12\log\det f'(C)
  \bigg\}
\end{align*}



\begin{align*}
  0&=C^*-f'(C)(C^*D^*+R^*R^*) \\
  0&=\big[\hat\beta_1f'(C)-\mu_1I+R\odot f''(C)\big](C-\tilde c)+f'(C)(R-\tilde r)
\end{align*}

\begin{align*}
  0&=-f'(q)+\frac12(C^*D^*+R^*R^*)^{-1}_{ij}\left(
    C^*_{jk}\frac{D^*_{ki}}{d_{01}}
    +
    2R^*_{jk}\frac{R^*_{ki}}{d_{01}}
    \right)
\end{align*}

\begin{equation}
  \hat\beta_2E_2-r_{22}^{(0)}\mu_2\frac12\left\{
    \hat\beta_2^2\big(f(1)-f(q_{22}^{(0)})\big)
    +\left(
      r_{12}^2+2\hat\beta_2r_{22}-\frac{2q_{12}r_{12}(r_{22}-r_{22}^{(0)})}{1-q_{22}^{(0)}}
    \right)\big(f'(1)-f'(q_{22}^{(0)})\big)
  \right\}
\end{equation}

What about the average for the Hessian terms?

\[
  \overline{
    |\det\operatorname{Hess}H(s_0)|\delta(\mu_0-\operatorname{Tr}\operatorname{Hess}H(s_0))|\det\operatorname{Hess}H(s_a)|\delta(\mu_1-\operatorname{Tr}\operatorname{Hess}H(s_a))
  }
\]


\paragraph{Acknowledgements}

\paragraph{Funding information}

\printbibliography

\end{document}
