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| -rw-r--r-- | when_annealed.tex | 3 |
1 files changed, 2 insertions, 1 deletions
diff --git a/when_annealed.tex b/when_annealed.tex index b07e3cd..5dc4065 100644 --- a/when_annealed.tex +++ b/when_annealed.tex @@ -110,6 +110,7 @@ product (or overlap) between the two configurations: \begin{equation} \label{eq:covariance} \overline{H(\pmb\sigma_1)H(\pmb\sigma_2)}=\frac1Nf\bigg(\frac{\pmb\sigma_1\cdot\pmb\sigma_2}N\bigg) \end{equation} +We will further take the distribution of $H$ to be centered, i.e., $\overline{H(\pmb\sigma)}=0$ for all $\pmb\sigma\in S^{N-1}$, which is equivalent to the absence of any deterministic term (or spike) in the function. Specifying the covariance function $f$ uniquely specifies the model. The series coefficients of $f$ need to be nonnnegative in order for $f$ to be a well-defined covariance. The case where $f$ is a homogeneous polynomial has @@ -130,7 +131,7 @@ called $3+s$ models.\footnote{ trivial overlap $q_0$ is also important in situations where a deterministic field (or spike) is present, as in \cite{Ros_2019_Complex}, but deterministic fields are likewise not considered here. -}These are examples of \emph{mixed} spherical models, which have been studied +} These are examples of \emph{mixed} spherical models, which have been studied in the physics and statistics literature and host a zoo of complex orders and phase transitions \cite{Crisanti_2004_Spherical, Crisanti_2006_Spherical, Krakoviack_2007_Comment, Crisanti_2007_Amorphous-amorphous, |