Other sources refer to the Hamiltonian that is the term inside the exponential function at 1 as free since it depends only on the increments between values. The immediate interpretation of 1 is that the values taken by the field at vertices which are close to each other are positively correlated. In some applications, this is the ideal intuition, but in others, it is more useful to think of the random walk bridge as a random height function indexed by [0,N], where the probability of a given path decomposes naturally into a product depending on the N increments, up to a normalising constant. Later, or in subsequent posts, we will heavily develop this idea. In particular, this means that the free field is Gaussian, and we can extract the covariances via where, under , is simple random walk started from x. Note that we can rearrange the Laplacian in 1 in terms of the transition kernel p of the simple random walk of D to obtain where is the transition matrix of SRW on D.

## Dgff. Order-Nr: DGFF4124500AN2P.

Other sources refer to the Hamiltonian that is the term inside the exponential function at 1 as free since it depends only on the increments between values. The immediate interpretation of 1 is that the values taken by the field at vertices which are close to each other are positively correlated. In some applications, this is the ideal intuition, but in others, it is more useful to think of the random walk bridge as a random height function indexed by [0,N], where the probability of a given path decomposes naturally into a product depending on the N increments, up to a normalising constant. Later, or in subsequent posts, we will heavily develop this idea. In particular, this means that the free field is Gaussian, and we can extract the covariances via where, under , is simple random walk started from x. Note that we can rearrange the Laplacian in 1 in terms of the transition kernel p of the simple random walk of D to obtain where is the transition matrix of SRW on D.

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