Langevin sampling of scalar fields

At each site integrate the scalar over the real line with a fixed flat measure. A conventional constant normalization per integration cancels from normalized expectations, but field-dependent Jacobians would not. The lattice action contains forward nearest-neighbor differences, a quadratic coefficient r, and a quartic coefficient u normalized by four factorial. All are dimensionless in these lattice units. This action describes a Euclidean or classical statistical field, not an oscillatory real-time path-integral weight.

For a finite periodic lattice, positive quartic coupling makes this polynomial action confining even when r is negative. In the purely Gaussian case, positive r gives a normalizable measure; setting both quartic coupling and r to zero leaves an unconstrained constant mode. A force calculation must include contributions from neighbors on both sides of each difference term. Incorrectly omitting one side can still produce smooth stochastic trajectories while sampling a different quadratic kernel. Physical dimensions and lattice-spacing factors will be restored explicitly in the continuum-scaling lesson.

Choose unit mobility in dimensionless stochastic time. The drift is minus the action gradient and the independent Wiener increments have variance equal to the time increment at each site. The square root of two multiplying the noise is essential: it sets the diffusion coefficient in the Fokker–Planck equation to one. Substituting the exponential of minus the action makes each probability-current component vanish, yielding the target stationary density when normalization and decay at field-space infinity justify the integration by parts.

This calculation establishes a stationary density for the exact diffusion. Convergence to it still requires appropriate ergodicity and regularity, and a finite trajectory retains warmup and correlation errors. Additive noise avoids an Itô-versus-Stratonovich drift ambiguity here; multiplicative noise requires a separate convention. A complex action cannot be inserted into the same real positive-density proof. Calling an algorithm Langevin does not establish correctness of complexified sampling or of a discretization that changes its noise covariance.

For one Gaussian mode with action r times field squared over two, Euler discretization gives an autoregressive process. Its coefficient is one minus epsilon times r and its fresh noise variance is twice epsilon. The variance remains bounded only when the coefficient magnitude is below one. Solving its stationary variance equation shows a step-size-dependent excess over the target inverse r. Running this stable chain longer estimates its biased variance more precisely rather than correcting the bias.

One remedy is a controlled extrapolation in stochastic step size, separately from spatial-spacing and finite-volume studies. A Metropolis-adjusted Langevin proposal instead includes the forward/reverse Gaussian proposal-density ratio; it cannot use only the symmetric-proposal acceptance rule. For a Gaussian mode the exact Ornstein–Uhlenbeck transition is another useful benchmark, though a generic interacting field has no such closed-form update. The heat instrument below shows deterministic mode decay only; it supplies no fluctuation noise and no equilibrium field samples.