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Jarek Duda (talk | contribs) →Basic model: Added Weighted MERW: Boltzmann path ensemble |
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Dividing by the probability of being at the <math>i</math>-th vertex, i.e. <math>\rho_i</math>, gives for the [[conditional probability]] <math>S_{ij}</math> of the <math>j</math>-th vertex being next after the <math>i</math>-th vertex
:<math>S_{ij} = \frac{A_{ij}}{\lambda} \frac{\psi_j}{\psi_i}</math>.
=== Weighted MERW: Boltzmann path ensemble ===
We have assumed that <math>A_{ij} \in \{0,1\} </math> for MERW corresponding to uniform ensemble among paths. However, the above derivation works for real nonnegative <math>A</math>. Parametrizing <math>A_{ij} = \exp(-E_{ij}) </math> and asking for probability of length <math>l </math> path <math>(\gamma_0, \ldots,\gamma_l) </math>, we get:
:<math>\textrm{Pr}(\gamma_0, \ldots,\gamma_l)=\rho_{\gamma_0} S_{\gamma_0 \gamma_1}\ldots S_{\gamma_{l-1}\gamma_l}= \psi_{\gamma_0} \frac{A_{\gamma_0 \gamma_1}\ldots A_{\gamma_{l-1}\gamma_l}}{\lambda^l} \psi_{\gamma_l}=\psi_{\gamma_0}\frac{\exp(-(E_{\gamma_0 \gamma_1}+\ldots +E_{\gamma_{l-1}\gamma_l}))}{\lambda^l} \psi_{\gamma_l} </math>
As in [[Boltzmann distribution]] of paths for energy defined as sum of <math>E_{ij} </math> over given path. For example it allows to calculate probability distribution of patterns in [[Ising model]].
== Examples ==
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