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Notes
- 1.
By pseudogap we mean a gap with zero width, i.e. the distribution is zero only in a point.
- 2.
When we say stable we mean that all the local stabilities are positive. In a thermodynamic sense those states are metastable.
- 3.
The magnetization is \(M=\sum _{\varvec{x}}^N s_{\varvec{x}}\).
- 4.
We say at least one, and not one and only one spin, because in principle the average number of triggered spins could be larger than one, and the avalanches stop due to the fluctuations in the number of triggered spins.
- 5.
The second of the two terms on the r.h.s. of Eq. 5.22 comes from Eq. 5.21. To find the first one it is necessary to calculate
$$\begin{aligned} \left\langle {\lambda (m)} \right\rangle _{m'} = \frac{\int _0^{\lambda (m')} \lambda \rho (\lambda )d\lambda }{\int _0^{\lambda (m')} \rho (\lambda )d\lambda },{(5.24)} \end{aligned}$$where the maximum stability of the chosen set, \(\lambda (m')\), can be evaluated through Eq. (5.17). Remembering that \(m'=2m\), one obtains \(\left\langle {\lambda (m)} \right\rangle _{m'}\sim \left( \frac{m}{N}\right) ^{\frac{1}{1+\theta }}\), that multiplied by m gives the term that appears in Eq. (5.22).
- 6.
We neglect the fluctuations of \(\sum _{\varvec{x}}\lambda _{\varvec{x}}\), since that sum is always positive and when m is large its fluctuations are small compared to its expectation value.
- 7.
In this chapter the averages \(\left\langle {\ldots } \right\rangle \) are averages over the avalanches.
- 8.
With at most logarithmic corrections, that can be neglected in this argument.
- 9.
It would be exactly the return probability of the random walk if the avalanche started with \(n_\mathrm {unst}=0\).
- 10.
We use an A, that stands for aleatory, because the R of random was already picked for the reluctant algorithm.
- 11.
The arguments of Sect. 5.3 for the scaling of \(\left\langle {\Delta M} \right\rangle \) and \(\left\langle {n} \right\rangle \) apply also to A and R dynamics. One obtains \(\left\langle {\Delta M} \right\rangle \sim \sqrt{N}\) for both the dynamics, \(\left\langle {n} \right\rangle \sim N\) for A and \(\left\langle {n} \right\rangle \sim N^{3/2}\) for R dynamics. Numerical simulations seem compatible with these trends in the limit of very large systems.
- 12.
In G avalanches \(n_\mathrm {unst}^*\) grows logarithmically, \(n_\mathrm {unst}^*\sim \log (N)\). With R dynamics we have little data because our measurements only go up to \(n_\mathrm {unst}=24\). We deduce a roughly linear scaling \(n_\mathrm {unst}^*\sim N\).
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Baity Jesi, M. (2016). Zero-Temperature Dynamics. In: Spin Glasses. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-41231-3_5
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