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| -rw-r--r-- | marginal.tex | 25 |
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diff --git a/marginal.tex b/marginal.tex index f19c096..ec28568 100644 --- a/marginal.tex +++ b/marginal.tex @@ -86,19 +86,23 @@ more useful. -An arbitrary function $g$ of the minimum eigenvalue of a matrix $A$ can be expressed as +An arbitrary function $g$ of the minimum eigenvalue of a matrix $A$ can be +expressed as \begin{equation} - g(\lambda_\textrm{min}(A)) - =\lim_{\beta\to\infty}\int\frac{d\mathbf x\,\delta(N-\mathbf x^T\mathbf x)e^{-\beta\mathbf x^TA\mathbf x}}{\int d\mathbf x'\,\delta(N-\mathbf x'^T\mathbf x')e^{-\beta\mathbf x'^TA\mathbf x'}}g\left(\frac{\mathbf x^TA\mathbf x}N\right) + g(\lambda_\textrm{max}(A)) + =\lim_{\beta\to\infty}\int + \frac{d\mathbf s\,\delta(N-\mathbf s^T\mathbf s)e^{\beta\mathbf s^TA\mathbf s}} + {\int d\mathbf s'\,\delta(N-\mathbf s'^T\mathbf s')e^{\beta\mathbf s'^TA\mathbf s'}} + g\left(\frac{\mathbf s^TA\mathbf s}N\right) \end{equation} Assuming \begin{equation} \begin{aligned} - &\lim_{\beta\to\infty}\int\frac{d\mathbf x\,\delta(N-\mathbf x^T\mathbf x)e^{-\beta\mathbf x^TA\mathbf x}}{\int d\mathbf x'\,\delta(N-\mathbf x'^T\mathbf x')e^{-\beta\mathbf x'^TA\mathbf x'}}g\left(\frac{\mathbf x^TA\mathbf x}N\right) \\ - &=\int\frac{d\mathbf x\,\delta(N-\mathbf x^T\mathbf x)\mathbb 1_{\operatorname{ker}(A-\lambda_\mathrm{min}(A)I)}(\mathbf x)}{\int d\mathbf x'\,\delta(N-\mathbf x'^T\mathbf x')\mathbb 1_{\operatorname{ker}(A-\lambda_\mathrm{min}(A)I)}(\mathbf x')}g\left(\frac{\mathbf x^TA\mathbf x}N\right) \\ - &=g(\lambda_\mathrm{min}(A)) - \frac{\int d\mathbf x\,\delta(N-\mathbf x^T\mathbf x)\mathbb 1_{\operatorname{ker}(A-\lambda_\mathrm{min}(A)I)}(\mathbf x)}{\int d\mathbf x'\,\delta(N-\mathbf x'^T\mathbf x')\mathbb 1_{\operatorname{ker}(A-\lambda_\mathrm{min}(A)I)}(\mathbf x')} \\ - &=g(\lambda_\mathrm{min}(A)) + &\lim_{\beta\to\infty}\int\frac{d\mathbf s\,\delta(N-\mathbf s^T\mathbf s)e^{\beta\mathbf s^TA\mathbf s}}{\int d\mathbf s'\,\delta(N-\mathbf s'^T\mathbf s')e^{\beta\mathbf s'^TA\mathbf s'}}g\left(\frac{\mathbf s^TA\mathbf s}N\right) \\ + &=\int\frac{d\mathbf s\,\delta(N-\mathbf s^T\mathbf s)\mathbb 1_{\operatorname{ker}(A-\lambda_\mathrm{max}(A)I)}(\mathbf s)}{\int d\mathbf s'\,\delta(N-\mathbf s'^T\mathbf s')\mathbb 1_{\operatorname{ker}(A-\lambda_\mathrm{max}(A)I)}(\mathbf s')}g\left(\frac{\mathbf s^TA\mathbf s}N\right) \\ + &=g(\lambda_\mathrm{max}(A)) + \frac{\int d\mathbf s\,\delta(N-\mathbf s^T\mathbf s)\mathbb 1_{\operatorname{ker}(A-\lambda_\mathrm{max}(A)I)}(\mathbf s)}{\int d\mathbf s'\,\delta(N-\mathbf s'^T\mathbf s')\mathbb 1_{\operatorname{ker}(A-\lambda_\mathrm{max}(A)I)}(\mathbf s')} \\ + &=g(\lambda_\mathrm{max}(A)) \end{aligned} \end{equation} The first relation extends a technique first introduced in @@ -108,9 +112,8 @@ over a spherical model whose Hamiltonian is quadratic with interaction matrix given by $A$. In the limit of zero temperature, the measure will concentrate on the ground states of the model, which correspond with the eigenspace of $A$ associated with its minimum eigenvalue $\lambda_\mathrm{min}$. The second -relation uses the fact that, once restricted to the sphere $\mathbf x^T\mathbf -x=N$ and the minimum eigenspace, $\mathbf x^TA\mathbf -x=N\lambda_\mathrm{min}(A)$. +relation uses the fact that, once restricted to the sphere $\mathbf s^T\mathbf +s=N$ and the minimum eigenspace, $\mathbf s^TA\mathbf s=N\lambda_\mathrm{min}(A)$. The relationship is formal, but we can make use of the fact that the integral expression with a Gibbs distribution can be manipulated with replica |