Reptile space

We emphasize that the reptile space for the electron paths depends on the proton coordinates so that, because of the fixed-node restriction for fermions, legal paths for S may or may not be legal for S and vice-versa. These two forms of importance sampling have the property that they sample regions of both configuration spaces (S and S ) and make the energy difference rigorously correct with a bounded variance. 

The first observation suggests that a good proposed move in MH from reptiles X to y should somehow linked them together using simple drift and diffusion moves. The most natural way to do this is to add /(< L) consecutive configurations generated by these moves. Such an addition also means that we must first remove some configurations from X to keep Y in the reptile space in addition,... 

Due to the high dimension and the complexity of T7, it is important to find an implementable and efficient sampling method to obtain reliable estimations. It is standard to use a Markov chain Monte Carlo (MCMC) method, in which a Markov chain is constructed on the space of reptiles (7 "( i)) with stationary distribution n. Under some regularity conditions, the chain converges to 17 and consequently, the reptiles simulated from the chain can be treated as an approximate sample from n. Samples of the pure and mixed distributions are then extracted from the middle and head/tail configurations of the reptiles to compute Monte Carlo estimates of various physical properties. 

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