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In contrast to GRW, the MERW transition probabilities generally depend on situation of the entire graph (are nonlocal). Hence, they rather should not be imagined as directly applied by the walker - if he performs randomly looking decisions based on local situation, like a person, the GRW approach is rather more appropriate. MERW is based on the [[principle of maximum entropy]], making it the safest assumption when we don't have any additional knowledge about the system. For example to model our knowledge about an object performing some complex dynamics - not necessarily random, like a particle.
=== Sketch of derivation ===
Assume for simplicity that the considered graph is indirected, connected, and aperiodic, what allows to conclude from [[Perron-Frobenius theorem]] that the dominant eigenvector is unique. Hence <math>A^l</math> can be asymptotically <math>(l\to\infty)</math> approximated with <math>\lambda^l \psi \psi^T</math> (or <math>\lambda^l |\psi\rangle \langle \psi|</math> in [[bra-ket notation]]).
MERW is uniform distribution among paths. The number of length <math>2l</math> paths with vertex <math>i</math> in the center is <math>\sum_{jk} (A^l)_{ji} (A^l)_{ik} </math> what asymptotically grows like <math>\lambda^{2l} \psi_i^2</math>, getting the <math>\rho_i\propto \psi_i^2</math> behavior. Analogously calculating probability distribution for two succeeding vertices, we can get the above formula for stochastic propagator <math>S_{ij}</math>.
== Examples ==
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