Markov processes on a circular lattice

Abstract: We develop a Markov process viewpoint for discrete circular distributions motivated by directional-statistics settings where angles are observed on a finite grid and evolve over time. On the $m$-point discrete circle, the cycle graph, we study diffusion-generated families, obtaining an explicit transition kernel, exact trigonometric moments, and convergence to uniformity. We present a simple approach to construct reversible nearest-neighbour chains with any prescribed strictly positive stationary pmf $π$, providing discrete analogues of Markov processes on the continuous circle. We construct processes whose stationary laws are the discrete von Mises and wrapped Cauchy distributions with closed-form normalizers and exact moments.

• The paper develops fractional-Laplacian diffusion semigroups on the cycle, providing Fourier-explicit transition kernels, exact trigonometric moment decay, and exponential convergence to uniformity at the spectral-gap rate.
• The paper constructs reversible nearest-neighbour chains for any strictly positive stationary distribution using rates proportional to square-root probability ratios, yielding discrete von Mises and wrapped Cauchy processes.
• The paper derives closed-form normalizers and moments for grid-aligned von Mises and wrapped Cauchy laws, while identifying open problems involving off-grid locations, sharper mixing bounds, and statistical inference.

Overview

This paper develops a Markov process framework for discrete circular distributions on the $m$-point circle, identified with the cycle graph $\mathbb{Z}_m$. The motivation comes from directional statistics settings where angles are observed on a finite grid—finite-resolution sensors, binned phase measurements, discretized bearings—and where observations are time-indexed, so that a model for temporal evolution is needed rather than a static pmf alone. The paper builds discrete analogues of two canonical continuous-circle constructions: diffusion-generated time-marginals (analogous to Brownian motion and the von Mises process of Kent) and drift-generated stationary laws (reversible nearest-neighbour chains targeting a prescribed equilibrium distribution). All constructions are accompanied by explicit transition kernels, closed-form normalizing constants, exact trigonometric moments, and quantitative convergence rates, which together support likelihood-based inference for time-varying discrete circular data.

Diffusion semigroups on the cycle

The first family of processes is generated by fractional powers of the combinatorial Laplacian $L$ on the cycle graph, defined by $(Lf)(r)=2f(r)-f(r+1)-f(r-1)$ with indices modulo $m$. For $\alpha>0$ and $\beta\in(0,1]$, the semigroup is

$P_t^{(\beta)} = \exp(-\alpha t L^\beta),$

where $L^\beta$ is defined via the spectral decomposition of the real symmetric matrix $L$. The case $\mathbb{Z}_m$0 recovers the standard heat semigroup (the continuous-time simple random walk on the cycle); $\mathbb{Z}_m$1 is a discrete analogue of the Poisson semigroup on the continuous circle. Because $\mathbb{Z}_m$2 and, for $\mathbb{Z}_m$3, $\mathbb{Z}_m$4 can be represented as a Bochner subordinate of the heat semigroup—preserving positivity—the kernel defines a valid continuous-time Markov chain on $\mathbb{Z}_m$5, lifted to an angle-valued process $\mathbb{Z}_m$6 on the grid $\mathbb{Z}_m$7.

The central structural result is an explicit Fourier representation of the transition kernel: since the characters $\mathbb{Z}_m$8 diagonalize $\mathbb{Z}_m$9 with eigenvalues $L$0, the kernel admits the series

$L$1

which depends only on $L$2, i.e., it is translation invariant. Uniformity is stationary, and every non-constant Fourier mode decays at rate $L$3, so any initial pmf converges to uniform as $L$4.

The eigenfunction structure yields exact trigonometric moment dynamics: each Fourier coefficient evolves multiplicatively, $L$5, equivalently $L$6. Starting from a point mass at $L$7, the mean direction is fixed at $L$8 and the first resultant length decays exactly as $L$9, giving a one-parameter concentration summary and a direct moment-matching estimator $(Lf)(r)=2f(r)-f(r+1)-f(r-1)$0.

Convergence to uniformity is quantified in total variation. With spectral gap $(Lf)(r)=2f(r)-f(r+1)-f(r-1)$1, the paper proves

$(Lf)(r)=2f(r)-f(r+1)-f(r-1)$2

via Parseval's identity applied to the mean-zero ratio $(Lf)(r)=2f(r)-f(r+1)-f(r-1)$3; from a point mass this becomes $(Lf)(r)=2f(r)-f(r+1)-f(r-1)$4. This bound is explicit but potentially loose: it uses only the worst-mode decay rate rather than mode-specific contributions, and no matching lower bound or cutoff phenomenon is established.

Reversible nearest-neighbour chains with prescribed stationary law

The complementary construction addresses the inverse problem: given any strictly positive target pmf $(Lf)(r)=2f(r)-f(r+1)-f(r-1)$5 on $(Lf)(r)=2f(r)-f(r+1)-f(r-1)$6, build a nearest-neighbour continuous-time Markov chain reversible with respect to $(Lf)(r)=2f(r)-f(r+1)-f(r-1)$7. The generator assigns rates

$(Lf)(r)=2f(r)-f(r+1)-f(r-1)$8

with $(Lf)(r)=2f(r)-f(r+1)-f(r-1)$9 set to make rows sum to zero. Detailed balance holds by construction ($m$0 for adjacent states), so $m$1 is the unique stationary distribution. When $m$2 is uniform this reduces to the standard constant-rate random walk; otherwise moves are biased toward higher-probability neighbours while retaining reversibility. This is the discrete counterpart of the mean-reverting circular diffusions characterized by Kent, and the construction is elementary yet fully general—it requires only strict positivity of $m$3.

Two specializations produce named processes:

Discrete von Mises process: taking $m$4 yields a reversible nearest-neighbour chain with that stationary law. When the location parameter lies on the grid, $m$5, the normalizer is independent of $m$6 and admits the closed form

$m$7

derived via the Fourier–Bessel expansion and a root-of-unity filter. Exact trigonometric moments follow: $m$8, which interpolates between the continuous von Mises moments and lattice effects through the aliasing index $m$9.

Discrete wrapped Cauchy process: taking $\alpha>0$0 proportional to Poisson kernel values $\alpha>0$1 again gives a reversible chain. For $\alpha>0$2 on the grid, the normalizer evaluates to $\alpha>0$3, yielding the explicit pmf

$\alpha>0$4

and, for $\alpha>0$5, the exact moment $\alpha>0$6. As $\alpha>0$7 these recover the continuous wrapped Cauchy characteristic function $\alpha>0$8, making the finite-$\alpha>0$9 correction term $\beta\in(0,1]$0 an explicit quantification of discretization bias.

Both derivations rest on the same root-of-unity filter technique, and both closed forms require the assumption that $\beta\in(0,1]$1 coincides with a grid point; off-grid location parameters leave the normalizers without the stated simplifications.

Limitations and open questions

The paper's guarantees are confined to the cycle graph with nearest-neighbour structure; extensions to other circulant graphs or higher-dimensional tori are not treated. The mixing bound is one-sided and coarse—a single exponential upper bound governed by the spectral gap—with no lower bounds, cutoff analysis, or comparison against the true total-variation decay. The closed-form normalizers and moments for the von Mises and wrapped Cauchy targets hold only when $\beta\in(0,1]$2 lies on the grid; the general off-grid case is left open. Finally, while the explicit kernels enable likelihood-based inference in principle, the paper contains no statistical analysis—no asymptotic theory for the proposed moment-matching estimator, no study of identifiability of $\beta\in(0,1]$3 jointly, and no application to the motivating datasets (roulette wheel, acrophase).

Conclusion

The paper supplies a complete, self-contained Markov process toolkit for the discrete circle: explicit fractional-Laplacian diffusion kernels with exact moment dynamics and TV convergence rates, and a universal Metropolis-type rate construction producing reversible nearest-neighbour chains for arbitrary positive targets, instantiated as discrete von Mises and wrapped Cauchy processes with closed-form normalizers and moments. The results bridge static discrete circular families of the kind catalogued by Mardia et al. with time-indexed modeling, and the remaining gaps—sharper mixing characterization, off-grid locations, and inferential theory—define the natural agenda for subsequent work.