Maximal entropy random walk: Difference between revisions

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Revision as of 03:29, 10 April 2025

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=== Weighted MERW: Boltzmann path ensemble ===

We have assumed that <math>A_{ij} \in \{0,1\} </math>, foryielding a MERW corresponding to the uniform ensemble among paths. However, the above derivation works for any real nonnegative <math>A</math> for which the Perron-Frobenius theorem applies. ParametrizingGiven <math>A_{ij} = \exp(-E_{ij}) </math>, and asking forthe probability of a particular length -<math>l </math> path <math>(\gamma_0, \ldots,\gamma_l) </math>, weis as getfollows:

:<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>,

Aswhich inis the same as the [[Boltzmann distribution]] of paths forwith energy defined as the sum of <math>E_{ij} </math> over giventhe edges of the path. For example, itthis allowscan be used with the transfer matrix to calculate the probability distribution of patterns in the [[Ising model]].

== Examples ==