Discrete Periodic Pitman Transform

• The discrete periodic Pitman transform is an algebraic-probabilistic method that uses coupled local update maps on periodic weight arrays to preserve polymer partition functions.
• It satisfies braid relations and underpins the invariance properties in both directed polymer and last-passage percolation models.
• The transform's Burke property and combinatorial characterizations ensure invariant measures and robust scaling limits in periodic integrable stochastic systems.

The discrete periodic Pitman transform is an algebraic and probabilistic transformation acting on collections of periodic weights in models such as directed polymers, last-passage percolation, and discrete integrable systems. Originating in work by Corwin, Gu, and collaborators, it generalizes the classical Pitman transform—which reflects a path S about its running maximum—to a setting where collections of weights (often organized as columns in a spatially periodic array) are transformed by coupled local update maps. The principal interest lies in its invariance properties, preservation of polymer partition functions, satisfaction of braid relations, and implications for invariant measures in periodic stochastic systems, including explicit combinatorial characterizations.

1. Formal Definition and Algebraic Structure

The discrete periodic Pitman transform operates on bi-infinite or periodic sequences of column-weight vectors. Given a weight array $W = (W_1, W_2, \dots, W_N)$ (each $W_k$ is a vector encoding the weights for column $k$), the transform focuses on pairs of adjacent columns and applies the coupled maps $T$ and $D$:

$$P_k(W_k, W_{k+1}) = (T(W_{k+1}, W_k), D(W_{k+1}, W_k)),$$

where $T$, $D$ are explicit (but model-dependent) maps designed such that polymer partition functions computed on the transformed weights are preserved. The action is local but—when extended—gives rise to global permutation invariance.

For directed polymers, the partition function between endpoints $(a,i)$ and $(c,j)$ is given by summing over up-right paths in the lattice and applying the corresponding weights:

$$Z^{\beta,W}(c,j\,|\,a,i) = \sum_{\pi\in\Pi_{(a,i),(c,j)}} \prod_{(r,\ell)\in\pi} e^{\beta W(r,\ell)}.$$

Matrix encodings via $H$-matrices (with $$H(W)_{i,j} = \sum_{i \leq \ell \leq j} e^{W_i + \cdots + W_\ell}$$ for $i \leq j$) allow the polymer observable to be written as a matrix product, and core invariance is expressed as:

$H(W_1) H(W_2) = H(T(W_2, W_1)) H(D(W_2, W_1)).$

This identity ensures partition function preservation under the discrete periodic Pitman transform (Engel et al., 7 Aug 2025).

2. Invariance Properties in Directed Polymer and Percolation Models

A principal result is the invariance of polymer partition functions under sequences of local transforms—specifically, permutations of the weight columns implemented via compositions of the $P_k$. For any endpoints $U$ and $V$ satisfying appropriate non-interference conditions, and any finite sequence of transforms,

$$Z^{P_{k_m}\cdots P_{k_1} W}(V\,|\,U) = Z^W(V\,|\,U).$$

This holds in both positive-temperature (e.g., directed polymers) and zero-temperature (last-passage percolation) regimes, with a max-plus version for the latter.

The mechanism relies crucially on the explicit algebraic encoding via the $H$-matrices and the local update property of $T, D$, guaranteeing that the multi-path observable is a global invariant under transformations generated by local operations. The result generalizes to multi-path partition functions, including joint observables of several families of up-right paths (Engel et al., 7 Aug 2025).

3. Braid Relations and Symmetric Group Action

Operators $P_k$ generating the transform satisfy the classical braid relations of the symmetric group:

$$P_k P_{k+1} P_k = P_{k+1} P_k P_{k+1}, \quad P_k^2 = \mathrm{Id}.$$

This symmetry ensures that the collection of all such operators (acting on all pairs of adjacent columns) constitute a representation of finite permutations of columns. The algebraic identities ensuring braid relations derive from properties of the local update maps and the $H$-matrix formalism—each braid relation corresponds to explicit matrix equalities. These properties are central in establishing equivalences between distinct orderings of update steps; algebraically, the transform provides a discrete, periodic version of the full-line Pitman transform's symmetric group invariance (Engel et al., 7 Aug 2025).

4. Burke Property and Invariance under Distributional Transformations

A salient probabilistic property—termed the Burke property—is that, for certain distributions of the weight environment (notably log–inverse–gamma, geometric, exponential), the discrete periodic Pitman transform preserves the joint law. Specifically, for independent sequences $X_1$, $X_2$ with the prescribed distributions,

$$(X_1, X_2) \stackrel{d}{=} (T(X_1, X_2), D(X_1, X_2)),$$

where $\stackrel{d}{=}$ denotes equality in distribution. This distributional invariance extends to permutations of the column parameters—under the action of $P_k$—which leaves the joint law of partition functions unchanged. The Burke property underpins multi-path invariance results for the periodic inverse–gamma polymer and ensures that fluctuation exponents for endpoint observables are robust to such columnwise re-orderings (Engel et al., 7 Aug 2025).

5. Combinatorial Characterization of Jointly Invariant Measures

In addition to algebraic and probabilistic invariance, the transform induces combinatorial structures in Markov chains defined on two–component vectors. For discrete-time Markov chains iterating the $D$ map,

$$(U_1^{(m+1)}, U_2^{(m+1)}) = (D(W, U_1^{(m)}), D(W, U_2^{(m)})),$$

the transformation $(X_1, X_2)\mapsto (X_1, D(X_1,X_2))$ is jointly invariant; its distribution is preserved by the chain (Engel et al., 7 Aug 2025).

Marginal measures on nonnegative vectors with fixed total mass are described combinatorially. If $X_1$ is uniform on vectors with total mass $\theta_1$, and $X_2$ on total mass $\theta_2$ ($\theta_1 < \theta_2$), the joint law of $(X_1, D(X_1, X_2))$ is supported on

$$\mathcal{Z}_{\theta_1, \theta_2} = \{ (u_1, u_2) : u_1(i) \leq u_2(i) \ \forall i \}.$$

The probability is given by

$$P(X_1 = u_1, D(X_1, X_2) = u_2) = \frac{\alpha(u_1, u_2)}{\binom{\theta_1 + N - 1}{N - 1} \binom{\theta_2 + N - 1}{N - 1}},$$

where $\alpha(u_1, u_2)$ is an explicit alternating product counting admissible pre-images for $u_2$ from $u_1$. This formula establishes a concrete link between the combinatorics of the transform and the distributional structure of invariant measures (Engel et al., 7 Aug 2025).

6. Implications for Periodic Stochastic Models and Scaling Limits

The discrete periodic Pitman transform is leveraged to analyze invariant measures and symmetries in periodic models for the KPZ equation, stochastic Burgers and stochastic heat equations, as well as semi-discrete directed polymer models. Variants of the transform are used to construct jointly invariant measures and to prove uniqueness theorems for long-time stationary distributions (Corwin et al., 5 Sep 2024). In positive and zero-temperature regimes, the combinatorial and probabilistic invariances persist under scaling limits—translating to the full-line model and connecting to KPZ universality phenomena.

A plausible implication is that this transform provides a robust algebraic–probabilistic tool that enables a unified treatment of invariance, symmetry, and explicit stationary distributions for periodic and infinite-dimensional stochastic systems.

7. Connections to Integrable Systems and Combinatorial Periodicity

The discrete periodic Pitman transform is closely related to integrable discrete systems, such as the box-ball system and ultra-discrete Toda lattice, where the Pitman-type reflection and associated shifts encode discrete periodicity and enable analysis of mass conservation, solitonic behavior, and invariant measures (Croydon et al., 2019, Croydon et al., 2020). The transformation lifts ideas from string combinatorics and periodicity interruption (Thierry, 2014), as well as links to transforms constructed from periodic bases (e.g., orthogonal complex conjugate periodic transforms (Shah et al., 2021)). The braid relations and invariant partition function properties further embed the transform in the algebraic integrability landscape.

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In summary, the discrete periodic Pitman transform is defined via local columnwise maps on periodic weight arrays; it preserves (multi-path) directed polymer partition functions, satisfies symmetric group braid relations, establishes the Burke property for a range of distributions, and enables combinatorial characterization of invariant measures. Its consequences extend to periodic stochastic PDEs, Markov chains, integrable systems, and scaling limits, making it an essential structure in algebraic probability and integrable combinatorics (Engel et al., 7 Aug 2025, Corwin et al., 5 Sep 2024).