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diff --git a/topology.tex b/topology.tex index 3e8ce9b..f420411 100644 --- a/topology.tex +++ b/topology.tex @@ -1,27 +1,19 @@ -\documentclass[a4paper,fleqn]{article} +\documentclass[submission, Phys]{SciPost} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} -\usepackage{amsmath,amssymb,latexsym,graphicx} -\usepackage{newtxtext,newtxmath} -\usepackage{bbold} +\usepackage{amsmath,latexsym,graphicx} +\usepackage[bitstream-charter]{mathdesign} \usepackage[dvipsnames]{xcolor} -\usepackage[ - colorlinks=true, - urlcolor=BlueViolet, - citecolor=BlueViolet, - filecolor=BlueViolet, - linkcolor=BlueViolet -]{hyperref} -\usepackage[ - style=phys, - eprint=true, - maxnames = 100 -]{biblatex} \usepackage{anyfontsize,authblk} -\usepackage{fullpage} -\addbibresource{topology.bib} +\hypersetup{ + colorlinks=true, + urlcolor={blue!50!black}, + citecolor={blue!50!black}, + filecolor={blue!50!black}, + linkcolor={blue!50!black} +} \title{ On the topology of solutions to random continuous constraint satisfaction problems @@ -39,15 +31,17 @@ independent Gaussian coefficients on the $(N-1)$-sphere. When solutions exist, they form a manifold. We compute the average Euler characteristic of this manifold in the limit of large $N$, and find different behavior - depending on the scaling of $M$ with $N$. When $\alpha=M/N$ is held constant, - the average characteristic is 2 whenever solutions exist. When $M$ is - constant, the average characteristic is also 2 up until a transition value - $M_\textrm{th}$, above which it is exponentially large in $N$. To better - interpret these results, we compute the average number of stationary points - of a test function on the solution manifold. In both regimes, this reveals - another transition between a regime with few and one with exponentially many - stationary points. We conjecture that this transition corresponds to a - geometric rather than a topological transition. + depending on the ratio $\alpha=M/N$. When $\alpha<\alpha_\text{onset}$, the + average characteristic is 2 and there is a single connected component, while + for $\alpha_\text{onset}<\alpha<\alpha_\text{shatter}$ a large connected + component coexists with many disconnected components. When + $\alpha>\alpha_\text{shatter}$ the large connected component vanishes, and the + entire manifold vanishes for $\alpha>\alpha_\text{\textsc{sat}}$. In the + limit $\alpha\to0$ there is a correspondence between this problem and the + topology of constant-energy level sets in the spherical spin glasses. We + conjecture that the energy $E_\text{shatter}$ associated with the vanishing of + the large connected component corresponds to the asymptotic limit of gradient + descent from a random initial condition. \end{abstract} \tableofcontents @@ -71,7 +65,12 @@ to solve them \cite{Baldassi_2016_Unreasonable, Baldassi_2019_Properties, Beneventano_2023_On}. Here, we show how \emph{topological} information about the set of solutions can be calculated in a simple model of satisfying random nonlinear equalities. This allows us to reason about the connectivity of this -solution set. +solution set. The topological properties revealed by this calculation yield +surprising results for the well-studied spherical spin glasses, where a +topological transition thought to occur at a threshold energy $E_\text{th}$ +where marginal minima are dominant is shown to occur at a different energy +$E_\text{shatter}$. We conjecture that this difference resolves an outstanding +problem in gradient descent dynamics in these systems. We consider the problem of finding configurations $\mathbf x\in\mathbb R^N$ lying on the $(N-1)$-sphere $\|\mathbf x\|^2=N$ that simultaneously satisfy $M$ @@ -80,20 +79,21 @@ constant $V_0\in\mathbb R$. The nonlinear constraints are taken to be centered Gaussian random functions with covariance \begin{equation} \label{eq:covariance} \overline{V_i(\mathbf x)V_j(\mathbf x')} - =\delta_{ij}F\left(\frac{\mathbf x\cdot\mathbf x'}N\right) + =\delta_{ij}f\left(\frac{\mathbf x\cdot\mathbf x'}N\right) \end{equation} -for some choice of function $F$. When the covariance function $F$ is polynomial, the -$V_k$ are also polynomial, with a term of degree $p$ in $F$ corresponding to +for some choice of function $f$. When the covariance function $f$ is polynomial, the +$V_k$ are also polynomial, with a term of degree $p$ in $f$ corresponding to all possible terms of degree $p$ in $V_k$. In particular, taking \begin{equation} V_k(\mathbf x) - =\sum_{p=0}^\infty\frac1{p!}\sqrt{\frac{F^{(p)}(0)}{N^p}} + =\sum_{p=0}^\infty\frac1{p!}\sqrt{\frac{f^{(p)}(0)}{N^p}} \sum_{i_1\cdots i_p}^NJ^{(k,p)}_{i_1\cdots i_p}x_{i_1}\cdots x_{i_p} \end{equation} with the elements of the tensors $J^{(k,p)}$ as independently distributed unit normal random variables satisfies \eqref{eq:covariance}. The size of the -series coefficients of $F$ therefore control the variances in the coefficients -of random polynomial constraints. +series coefficients of $f$ therefore control the variances in the coefficients +of random polynomial constraints. When $M=1$, this problem corresponds to the +level set of a spherical spin glass with energy density $E=\sqrt{N}V_0$. This problem or small variations thereof have attracted attention recently for @@ -112,21 +112,94 @@ directly. This set can be written as \begin{equation} - \Omega=\big\{\mathbf x\in\mathbb R^N\mid \|\mathbf x\|^2=N,V_k(\mathbf x)=0 + \Omega=\big\{\mathbf x\in\mathbb R^N\mid \|\mathbf x\|^2=N,V_k(\mathbf x)=V_0 \;\forall\;k=1,\ldots,M\big\} \end{equation} -Because the constraints are all smooth functions, $\Omega$ is almost always a manifold without singular points. The conditions for a singular point are that +Because the constraints are all smooth functions, $\Omega$ is almost always a manifold without singular points.\footnote{The conditions for a singular point are that $0=\frac\partial{\partial\mathbf x}V_k(\mathbf x)$ for all $k$. This is equivalent to asking that the constraints $V_k$ all have a stationary point at the same place. When the $V_k$ are independent and random, this is vanishingly unlikely, requiring $NM+1$ independent equations to be simultaneously satisfied. This means that different connected components of the set of solutions do not -intersect, nor are there self-intersections, without extraordinary fine-tuning. +intersect, nor are there self-intersections, without extraordinary fine-tuning.} +We study the topology of the manifold $\Omega$ by two related means: its +average Euler characteristic, and the average number of stationary points of a +linear height function restricted to the manifold. These measures tell us +complementary pieces of information, respectively the alternating sum and +direct sum of the Betti numbers of $\Omega$. We find that for the varied cases +we study, these two always coincide at the largest exponential order in $N$, +putting strong constraints on the resulting topology and geometry. + +\section{Results} -When $M$ is too large, no solutions exist and $\Omega$ becomes the empty set. -Following previous work, a replica symmetric equilibrium calculation using the -cost function \eqref{eq:cost} predicts that solutions vanish when the ratio -$\alpha=M/N$ is larger than $\alpha_\text{\textsc{sat}}=f'(1)/f(1)$. Based on the results of this paper, and the fact that this $\alpha_\text{\textsc{sat}}$ is consistent +\subsection{Topology of solutions to many equations and the satisfiability transition} + +\begin{figure} + \includegraphics{figs/phases_1.pdf} + \hspace{-3em} + \includegraphics{figs/phases_2.pdf} + \hspace{-3em} + \includegraphics{figs/phases_3.pdf} + + \caption{ + Topological phases of the model for three different homogeneous covariance + functions. The onset transition $V_\text{onset}$, shattering transition + $V_\text{shatter}$, and satisfiability transition $V_\text{\textsc{sat}}$ + are indicated when they exist. In the limit of $\alpha\to0$, the behavior + of level sets of the spherical spin glasses are recovered: the final plot + shows how in the pure cubic model the ground state energy $E_\text{gs}$ and threshold energy + $E_\text{th}$ correspond with the limits of the satisfiability and + shattering transitions, respectively. Note that for mixed models with + inhomogeneous covariance functions, $E_\text{th}$ is not the lower limit of + $V_\text{sh}$. + } +\end{figure} + +\begin{figure} + \includegraphics[width=0.24\textwidth]{figs/connected.pdf} + \hfill + \includegraphics[width=0.24\textwidth]{figs/coexist.pdf} + \hfill + \includegraphics[width=0.24\textwidth]{figs/shattered.pdf} + \hfill + \includegraphics[width=0.24\textwidth]{figs/gone.pdf} + + \includegraphics{figs/bar.pdf} + + \caption{ + Cartoon of the topology of the solution manifold implied by our + calculation. The arrow shows the vector $\mathbf x_0$ defining the height + function. The region of solutions is marked in black, and the critical points + of the height function restricted to this region are marked with a point. + For $\alpha<1$, there are few simply connected regions with most of the + minima and maxima contributing to the Euler characteristic concentrated at + the height $m^*$. For $\alpha\geq1$, there are many simply + connected regions and most of their minima and maxima are concentrated at + the equator. + } \label{fig:cartoons} +\end{figure} + +\subsection{Topology of level sets of the spherical spin glasses and the dynamic threshold} + +\begin{figure} + \includegraphics{figs/dynamics_2.pdf} + \hspace{-0.5em} + \includegraphics{figs/dynamics_3.pdf} + + \caption{ + Comparison of the shattering energy $E_\text{sh}$ with the asymptotic + performance of gradient descent from a random initial condition in $p+s$ + models with $p=2$ and $p=3$ and varying $s$. The values of $\lambda$ depend on $p$ and $s$ and are taken from \cite{Folena_2023_On}. The points show the asymptotic performance + extrapolated using two different methods and have unknown uncertainty, from \cite{Folena_2023_On}. Also + shown is the annealed threshold energy $E_\text{th}$, where marginal minima + are the most common type of stationary point. The section of $E_\text{sh}$ + that is dashed on the left plot indicates the continuation of the annealed + result, whereas the solid portion gives the value calculated with a + {\oldstylenums 1}\textsc{frsb} ansatz. + } +\end{figure} + +\section{The average Euler characteristic} The Euler characteristic $\chi$ of a manifold is a topological invariant \cite{Hatcher_2002_Algebraic}. It is perhaps most familiar in the context of connected compact orientable surfaces, where it @@ -153,7 +226,7 @@ points. Since the sign of the determinant of the Hessian matrix of $H$ at a stationary point is equal to its index, if we count stationary points including the sign of the determinant, we arrive at the Euler characteristic, or \begin{equation} \label{eq:kac-rice} - \chi=\int_\Omega d\mathbf x\,\delta\big(\nabla H(\mathbf x)\big)\det\operatorname{Hess}H(\mathbf x) + \chi(\Omega)=\int_\Omega d\mathbf x\,\delta\big(\nabla H(\mathbf x)\big)\det\operatorname{Hess}H(\mathbf x) \end{equation} When the Kac--Rice formula is used to \emph{count} stationary points, the sign of the determinant is a nuisance that one must take pains to preserve @@ -167,8 +240,6 @@ x)=\mathbf x_0\cdot\mathbf x$ for some $\mathbf x_0\in\mathbb R^N$ with $\|\mathbf x_0\|^2=N$. $H$ is a height function because when $\mathbf x_0$ is used as the polar axis, $H$ gives the height on the sphere. -\section{The average Euler characteristic} - We treat the integral over the implicitly defined manifold $\Omega$ using the method of Lagrange multipliers. We introduce one multiplier $\omega_0$ to enforce the spherical constraint and $M$ multipliers $\omega_k$ to enforce the vanishing of @@ -176,7 +247,7 @@ each of the $V_k$, resulting in the Lagrangian \begin{equation} L(\mathbf x,\pmb\omega) =H(\mathbf x)+\frac12\omega_0\big(\|\mathbf x\|^2-N\big) - +\sum_{k=1}^M\omega_kV_k(\mathbf x) + +\sum_{k=1}^M\omega_k\big(V_k(\mathbf x)-V_0\big) \end{equation} The integral over $\Omega$ in \eqref{eq:kac-rice} then becomes \begin{equation} \label{eq:kac-rice.lagrange} @@ -202,11 +273,11 @@ odd-dimensional manifolds. This is the signature of it in this problem. To evaluate the average of $\chi$ over the constraints, we first translate the $\delta$ functions and determinant to integral form, with \begin{align} \delta\big(\partial L(\mathbf x,\pmb\omega)\big) - =\int\frac{d\hat{\mathbf x}}{(2\pi)^N}\frac{d\hat{\pmb\omega}}{(2\pi)^{M+1}} + &=\int\frac{d\hat{\mathbf x}}{(2\pi)^N}\frac{d\hat{\pmb\omega}}{(2\pi)^{M+1}} e^{i[\hat{\mathbf x},\hat{\pmb\omega}]\cdot\partial L(\mathbf x,\pmb\omega)} \\ \det\partial\partial L(\mathbf x,\pmb\omega) - =\int d\bar{\pmb\eta}\,d\pmb\eta\,d\bar{\pmb\gamma}\,d\pmb\gamma\, + &=\int d\bar{\pmb\eta}\,d\pmb\eta\,d\bar{\pmb\gamma}\,d\pmb\gamma\, e^{-[\bar{\pmb\eta},\bar{\pmb\gamma}]^T\partial\partial L(\mathbf x,\pmb\omega)[\pmb\eta,\pmb\gamma]} \end{align} @@ -240,7 +311,7 @@ Since this is an exponential integrand linear in the functions $V_k$, we can ave +\frac{i}2\sigma_0(1)\big(\|\pmb\phi(1)\|^2-N\big) -iV_0\sum_{k=1}^M\sigma_k(1) \right] \\ - -\frac12\int d1\,d2\,\sum_{k=1}^M\sigma_k(1)\sigma_k(2)F\left(\frac{\pmb\phi(1)\cdot\pmb\phi(2)}N\right) + -\frac12\int d1\,d2\,\sum_{k=1}^M\sigma_k(1)\sigma_k(2)f\left(\frac{\pmb\phi(1)\cdot\pmb\phi(2)}N\right) \Bigg\} \end{aligned} \end{equation} @@ -254,8 +325,8 @@ Performing that integral yields H(\pmb\phi(1)) +\frac{i}2\sigma_0(1)\big(\|\pmb\phi(1)\|^2-N\big) \right] \\ - &\hspace{5em}-\frac M2V_0^2\int d1\,d2\,F\left(\frac{\pmb\phi(1)\cdot\pmb\phi(2)}N\right)^{-1} - -\frac M2\log\operatorname{sdet}F\left(\frac{\pmb\phi(1)\cdot\pmb\phi(2)}N\right) + &\hspace{5em}-\frac M2V_0^2\int d1\,d2\,f\left(\frac{\pmb\phi(1)\cdot\pmb\phi(2)}N\right)^{-1} + -\frac M2\log\operatorname{sdet}f\left(\frac{\pmb\phi(1)\cdot\pmb\phi(2)}N\right) \Bigg\} \end{aligned} \end{equation} @@ -278,8 +349,8 @@ These new variables can replace $\pmb\phi$ in the integral using a generalized H \mathbb M(1) +\frac{i}2\sigma_0(1)\big(\mathbb Q(1,1)-1\big) \right] \\ - &\hspace{5em}-\frac M2V_0^2\int d1\,d2\,F(\mathbb Q)^{-1}(1,2) - -\frac M2\log\operatorname{sdet}F(\mathbb Q) + &\hspace{5em}-\frac M2V_0^2\int d1\,d2\,f(\mathbb Q)^{-1}(1,2) + -\frac M2\log\operatorname{sdet}f(\mathbb Q) +\frac N2\log\operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T) \Bigg\} \end{aligned} @@ -339,8 +410,7 @@ We can treat the integral over $\sigma_0$ immediately. It gives =2\pi\,\delta(C-1)\,\delta(G+R)\,\bar HH \end{equation} This therefore sets $C=1$ and $G=-R$ in the remainder of the integrand, as well -as setting everything depending on $\bar H$ and $H$ to zero. - +as setting everything depending on $\bar H$ and $H$ to zero. With these solutions inserted, the remaining remaining terms in the exponential give \begin{equation} \begin{aligned} \operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T) @@ -360,24 +430,59 @@ as setting everything depending on $\bar H$ and $H$ to zero. +\frac{2f(1)}{R^3f'(1)}\bar{\hat H}\hat H \end{equation} \begin{equation} - \int d1\,d2\,F(\mathbb Q)^{-1}(1,2) + \int d1\,d2\,f(\mathbb Q)^{-1}(1,2) =\left(f(1)+\frac{R^2f'(1)}{D}\right)^{-1} +2\frac{Rf'(1)}{(Df(1)+R^2f'(1))^2}\bar{\hat H}\hat H \end{equation} +\begin{equation} + D=-\frac{m+R}{1-m^2} \qquad \hat m=0 +\end{equation} +\begin{equation} + \mathcal S(R,D,m,\hat m\mid\alpha,V_0) + =\hat m-\frac\alpha2\left[ + \log\left(1+\frac{f(1)D}{f'(1)R^2}\right) + +V_0^2\left(f(1)+\frac{f'(1)R^2}{D}\right)^{-1} + \right] + +\frac12\log\left( + 1+\frac{(1-m^2)D+\hat m^2-2Rm\hat m}{R^2} + \right) +\end{equation} +When $\alpha<\alpha_\text{onset}$ this potential has maxima at $\pm m^*$ with +$m^*=-R^*$ where its value is zero. +\begin{equation} + \alpha_\text{onset}=1-\left(\frac{V_0^2}{V_0^2+f(1)}\right)^2 +\end{equation} +\begin{equation} + \alpha_\text{shatter}=4V_0^2f(1)f'(1)\frac{f'(1)-f(1)}{\big((V_0^2+f(1))f'(1)-f(1)^2\big)^2} +\end{equation} -\subsection{Behavior with extensively many constraints} - -\subsection{Behavior with finitely many constraints} - -The correct scaling to find a nontrivial answer with finite $M$ is to scale -both the covariance functions and fixed constants with $N$ like -$v_0=\frac1NV_0$, $f(q)=\frac1NF(q)$, so that $v_0$ and $f(q)$ are finite at -large $N$. With these scalings and $M=1$, this problem reduces to examining the -levels sets of the spherical spin glasses at energy density $E=v_0$. +\section{Complexity of the height function} -$v_0^{\chi>2}=\sqrt{2f(1)}$ +\section{Implications for the spherical spin glasses} -$v_0^{m=0}=2\sqrt{f(1)-\frac{f(1)^2}{f'(1)}}$ +As indicated earlier, for $M=1$ the solution manifold corresponds to the energy +level set of a spherical spin glass with energy density $E=\sqrt NV_0$. All the +results from the previous sections follow, and can be translated to the spin +glasses by taking the limit $\alpha\to0$ while scaling $V_0=\sqrt\alpha E$. With a little algebra this procedure yields +\begin{equation} + E_\text{onset}=\pm\sqrt{2f(1)} +\end{equation} +\begin{equation} + E_\text{shatter}=\pm\sqrt{4f(1)\left(1-\frac{f(1)}{f'(1)}\right)} +\end{equation} +for the energies at which level sets of the spherical spin glasses have +disconnected pieces appear, and that at which a large connected component +vanishes. For the pure $p$-spin spherical spin glasses with $f(q)=\frac12q^p$, +$E_\text{shatter}=\sqrt{2(p-1)/p}$, precisely the threshold energy in these +models. This is expected, since threshold energy, defined as the place where +marginal minima are dominant in the landscape, is widely understood as the +place where level sets are broken into pieces. + +However, for general mixed models the threshold energy is +\begin{equation} + E_\mathrm{th}=\pm\frac{f(1)[f''(1)-f'(1)]+f'(1)^2}{f'(1)\sqrt{f''(1)}} +\end{equation} +which generally satisfies $|E_\text{shatter}|\leq|E_\text{th}|$. \subsection{What does the average Euler characteristic tell us?} @@ -426,27 +531,6 @@ Thus we find the average Euler characteristic in this simple example is 2 despite the fact that the possible manifolds resulting from the constraints have characteristics of either 0 or 4. -\begin{figure} - \includegraphics[width=0.32\columnwidth]{figs/connected.pdf} - \hfill - \includegraphics[width=0.32\columnwidth]{figs/shattered.pdf} - \hfill - \includegraphics[width=0.32\columnwidth]{figs/gone.pdf} - - \includegraphics{figs/bar.pdf} - - \caption{ - Cartoon of the topology of the solution manifold implied by our - calculation. The arrow shows the vector $\mathbf x_0$ defining the height - function. The region of solutions is marked in black, and the critical points - of the height function restricted to this region are marked with a point. - For $\alpha<1$, there are few simply connected regions with most of the - minima and maxima contributing to the Euler characteristic concentrated at - the height $m^*$. For $\alpha\geq1$, there are many simply - connected regions and most of their minima and maxima are concentrated at - the equator. - } \label{fig:cartoons} -\end{figure} \cite{Franz_2016_The, Franz_2017_Universality, Franz_2019_Critical, Annesi_2023_Star-shaped, Baldassi_2023_Typical} @@ -471,58 +555,89 @@ have characteristics of either 0 or 4. \section{Interpretation of our results} -\paragraph{Quenched average of the Euler characteristic.} - -\begin{equation} - D=\beta R - \qquad - \hat\beta=-\frac{m+\sum_aR_{1a}}{\sum_aC_{1a}} +\[ + E=\frac1{\hat\omega_1}\frac12\left[ + \hat\omega_1^2f(C)+(2\omega_1\hat\omega_1R-\omega_1^2D)f'(C)+\omega_1^2(R^2-G^2)f''(C)+\log\det\frac{CD+R^2}{G^2} + \right] +\] +$D=\beta R$, $R=r_dI$, $G=-r_dI$ +\[ + E=\frac1{\hat\omega_1}\frac12\left[ + \hat\omega_1^2f(C)+(2\omega_1\hat\omega_1-\omega_1^2\beta)r_df'(1)+\log\det(\beta r_d^{-1}C+I) + \right] +\] +$\beta=\hat\omega_1/\omega_1$ +\[ + E=\frac1{\hat\omega_1}\frac12\left[ + \hat\omega_1^2f(C)+\omega_1\hat\omega_1r_df'(1)+\log\det(\hat\omega_1\omega_1^{-1}r_d^{-1}C+I) + \right] +\] +$z=r_d\omega_1$ +\[ + E=\frac1{\hat\omega_1}\frac12\left[ + \hat\omega_1^2f(C)+\hat \omega_1 zf'(1)+\log\det(\hat\omega_1C/z+I) + \right] +\] +\[ + \log\chi + =-\hat\omega_1 E + +\frac12[2\omega_1\hat\omega_1r_d-\omega_1^2d_d]f'(1) + +\frac12\int_0^1dq\,\left[ + \hat\omega_1^2f''(q)\chi(q) + +\frac1{\chi(q)+r_d^2/d_d} + \right] + -\frac12\log r_d^2 +\] +\[ + 0=\frac12\hat\omega_1^2f''(q)-\frac12\frac1{[\chi(q)+r_d^2/d_d]^2} +\] +\[ + \chi_0(q)=\frac1{\hat\omega_1}f''(q)^{-1/2}-\frac{r_d^2}{d_d} +\] + +\[ + \chi(q)=\begin{cases} + \chi_0(q) & q < q_0 \\ + 1-(1-m)q_1-mq & q_0 < q < q_1 \\ + 1-q & q > q_1 + \end{cases} +\] +\[ + 0=\hat\omega_1r_d-\omega_1d_d \qquad - \hat m=0 -\end{equation} - -\begin{align} - &\mathcal S(m,C,R) - =\frac12\log\det\big[I+\hat\beta R^{-1}(C-m^2)\big] \notag \\ - &\quad-\frac\alpha2\log\det\big[I+\hat\beta\big(R\odot f'(C)\big)^{-1}f(C)\big] -\end{align} - -The quenched average of the Euler characteristic in the replica symmetric ansatz becomes for $1<\alpha<\alpha_\text{\textsc{sat}}$ -\begin{align} - \frac1N\overline{\log\chi} - =\frac12\bigg[ - \log\left(-\frac 1{\tilde r_d}\right) - -\alpha\log\left( - 1-\Delta f\frac{1+\tilde r_d}{f'(1)\tilde r_d} - \right) \notag \\ - -\alpha f(0)\left(\Delta f-\frac{f'(1)\tilde r_d}{1+\tilde r_d}\right)^{-1} - \bigg] -\end{align} -where $\Delta f=f(1)-f(0)$ and $\tilde r_d$ is given by -\begin{align} - \tilde r_d - =-\frac{f'(1)f(1)-\Delta f^2}{2(f'(1)-\Delta f)^2} - \bigg( - \alpha-2+\frac{2f'(1)f(0)}{f'(1)f(1)-\Delta f^2} \notag\\ - +\sqrt{ - \alpha^2 - -4\alpha\frac{f'(1)f(0)\Delta f\big(f'(1)-\Delta f\big)}{\big(f'(1)f(1)-\Delta f^2\big)^2} - } - \bigg) -\end{align} -When $\alpha\to\alpha_\text{\textsc{sat}}=f'(1)/f(1)$ from below, $\tilde r_d\to -1$, which produces $N^{-1}\overline{\log\chi}\to0$. + \omega_1=\hat\omega_1\frac{r_d}{d_d} +\] +\[ + \log\chi + =-\hat\omega_1 E + +\frac12\hat\omega_1^2r_d^2/d_df'(1) + +\frac12\int_0^1dq\,\left[ + \hat\omega_1^2f''(q)\chi(q) + +\frac1{\chi(q)+r_d^2/d_d} + \right] + -\frac12\log r_d^2 +\] +\[ + 0=-\frac{\hat\omega_1^2f'(1)}{d_d}+\int_0^1dq\,\frac1{(r_d^2/d_d+\chi(q))^2} +\] +\[ + d_d=-\frac{1+r_d}{\int dq\,\chi(q)}r_d +\] + +\paragraph{Acknowledgements} +The authors thank Pierfrancesco Urbani for helpful conversations on these topics. -\section*{Acknowledgements} -\addcontentsline{toc}{section}{Acknowledgements} +\paragraph{Funding information} JK-D is supported by a \textsc{DynSysMath} Specific Initiative of the INFN. -The authors thank Pierfrancesco Urbani for helpful conversations on these topics. \appendix \section{Calculation of the prefactor of the average Euler characteristic} \label{sec:prefactor} -\printbibliography -\addcontentsline{toc}{section}{References} +\section{The quenched shattering energy in {\oldstylenums 1}\textsc{frsb} models} +\label{sec:1frsb} + +\bibliography{topology} \end{document} |