Burnside Process in Markov Chains

• Burnside Process is a Markov chain based on finite group actions that samples orbits uniformly using stabilizers and fixed-point sets.
• It connects group theory with combinatorial structures such as set partitions, contingency tables, and parking functions through explicit sampling techniques.
• The process leverages spectral analysis and orthogonal polynomials to achieve rapid mixing, with quantitative convergence bounds validating its efficiency.

Searching arXiv for papers on the Burnside process and related variants. The Burnside process is a Markov chain attached to a finite group action $G\curvearrowright X$. From a current state $x\in X$, it samples a group element from the stabilizer of $x$ and then samples a new state from the corresponding fixed-point set, producing a reversible chain on $X$ whose projection to the orbit space $X/G$ has the uniform stationary distribution on orbits. In this form it is a general orbit-sampling mechanism rather than a sampler on orbit labels directly, and it has been developed for set partitions, conjugacy classes, contingency tables, parking functions, Dyck paths, and several weighted or dual variants (Feng, 29 Oct 2025, Paguyo, 2022).

1. General construction and stationary law

Let $X$ be a finite set and let $G$ be a finite group acting on $X$. For $x\in X$, write $G_x=\{g\in G:g\cdot x=x\}$ for the stabilizer, $x\in X$0 for the fixed-point set, and $x\in X$1 for the orbit of $x\in X$2. The classical Burnside kernel is

$x\in X$3

Equivalently, one step is: choose $x\in X$4 uniformly, then choose $x\in X$5 uniformly. A direct detailed-balance calculation gives the stationary law

$x\in X$6

where $x\in X$7. Burnside’s lemma,

$x\in X$8

provides the normalization. When the chain is lumped by orbits, the induced chain on $x\in X$9 has the uniform stationary distribution. Since the identity lies in every stabilizer and fixes every state, all entries $x$0 are positive, so the chain is irreducible and aperiodic (Feng, 29 Oct 2025).

In the terminology of Diaconis and Zhong, the Burnside process is also a special case of the Swendsen–Wang or data-augmentation paradigm. In the examples where the lumped chain admits an explicit diagonalization, this interpretation connects orbit sampling to orthogonal-polynomial spectral theory, beta-binomial mixtures, and Mallows-type distributions on permutations (Diaconis et al., 2020).

2. Value permutations on $x$1 and uniform sampling of set partitions

A central model takes $x$2 with $x$3, and $x$4 acting by permuting the values of a tuple coordinate-wise: $x$5 If $x$6, let $x$7 be the number of distinct values appearing in $x$8, and define $x$9 by $X$0 iff $X$1. Then two tuples lie in the same orbit exactly when they induce the same set partition of $X$2, and

$X$3

Thus the orbit space is identified with set partitions of $X$4, and running the Burnside chain on $X$5 and returning $X$6 yields a sampler for the uniform distribution on $X$7. In this model the kernel admits a closed form: if $X$8, $X$9 and $X/G$0 are their numbers of distinct values, and $X/G$1 is the size of the union of their value-sets, then

$X/G$2

where $X/G$3 is the number of fixed points of a uniformly random permutation in $X/G$4. The chain is reversible with

$X/G$5

so that the lumped chain on $X/G$6 has the uniform stationary distribution (Paguyo, 2022).

The same work gives quantitative convergence bounds. For $X/G$7, if

$X/G$8

then

$X/G$9

The proof uses a two-stage coupling: first coupling stabilizer permutations on the union of the free labels, then coupling the relabeling step by a maximal-matching coupling on fixed-point sets. For $X$0, a global minorization gives

$X$1

for all $X$2, hence

$X$3

On the lumped chain $X$4 on $X$5, the spectral gap satisfies

$X$6

so

$X$7

up to polynomial-in-$X$8 factors. The resulting sampler is therefore rapidly mixing when $X$9 grows at most polynomially in $G$0 (Paguyo, 2022).

3. Coordinate permutations, the binary cube, and Hahn-polynomial diagonalization

A second explicit setting uses the action of $G$1 on $G$2 by permuting coordinates. All binary strings of Hamming weight $G$3 lie in a single orbit, so the Burnside chain lumps to a chain $G$4 on $G$5 recording the current weight. From a binary vector $G$6 of weight $G$7, one samples a permutation $G$8 uniformly from the stabilizer of $G$9, writes $X$0 in cycle form, independently labels each cycle by $X$1 or $X$2 with probability $X$3, and assigns that label to all coordinates in the cycle. The lumped chain is reversible with the uniform law

$X$4

This provides one of the cleanest exact Burnside-process spectral analyses (Diaconis et al., 2020).

The orthogonal polynomials for the uniform weight on $X$5 are the Hahn polynomials with parameters $X$6, which here become the discrete Chebyshev polynomials $X$7. The spectrum is explicit: all odd-degree polynomials $X$8 are annihilated, while the even-degree polynomials $X$9 are eigenfunctions with eigenvalues

$x\in X$0

The $x\in X$1-step kernel therefore has a full spectral expansion in the $x\in X$2. Using orthogonality and a $x\in X$3-bound, one gets

$x\in X$4

for worst-case initial states such as $x\in X$5 or $x\in X$6, together with the matching lower bound

$x\in X$7

Hence the chain mixes in exactly $x\in X$8 steps. This example is the setting in which the Burnside process is diagonalized by Hahn polynomials and where sharp geometric convergence is obtained rather than only polynomial upper bounds (Diaconis et al., 2020).

4. Partitions, contingency tables, parking functions, and Dyck paths

When $x\in X$9 and the action is conjugation, the orbits are conjugacy classes, equivalently integer partitions $G_x=\{g\in G:g\cdot x=x\}$0. The Burnside step becomes: from a permutation $G_x=\{g\in G:g\cdot x=x\}$1, sample $G_x=\{g\in G:g\cdot x=x\}$2 uniformly from the centralizer $G_x=\{g\in G:g\cdot x=x\}$3, then sample $G_x=\{g\in G:g\cdot x=x\}$4 uniformly from $G_x=\{g\in G:g\cdot x=x\}$5, and set the next state to $G_x=\{g\in G:g\cdot x=x\}$6. On cycle counts $G_x=\{g\in G:g\cdot x=x\}$7, the centralizer decomposes as

$G_x=\{g\in G:g\cdot x=x\}$8

and the cycle-type of a uniform element in each $G_x=\{g\in G:g\cdot x=x\}$9 can be sampled by a discrete stick-breaking construction together with independent uniform choices in $x\in X$00. The resulting lumped Burnside step on partitions runs in average $x\in X$01 time and space. A parallel construction for the action of $x\in X$02 on $x\in X$03 by left-right multiplication yields a Burnside sampler for contingency tables with margins $x\in X$04; in lumped form it runs in $x\in X$05 average time and memory and exploits Fisher–Yates sampling on cellwise cycle data (Diaconis et al., 4 Mar 2025).

For Catalan structures, let $x\in X$06 be the parking functions and let $x\in X$07 act by permuting coordinates. Because the parking condition depends only on the sorted multiset of entries, orbits are indexed by weakly increasing parking functions $x\in X$08, and $x\in X$09. The stationary distribution on $x\in X$10 is

$x\in X$11

where $x\in X$12 is the histogram of value $x\in X$13, so the lumped chain is uniform on $x\in X$14. Via the $x\in X$15-equivariant bijection $x\in X$16, the same