Periodic Inverse-Gamma Polymer

• Periodic inverse-gamma polymer is a lattice model with cyclic log-inverse-gamma distributed weights introducing novel algebraic invariances and integrability.
• The discrete periodic Pitman transform preserves multi-path partition functions by satisfying braid relations, linking the model to the geometric RSK correspondence.
• The model exhibits KPZ universality with characteristic scaling exponents and explicit invariant measures, paving the way for deeper analysis of periodic and integrable systems.

A periodic inverse-gamma polymer is a directed polymer model in a random environment on a two-dimensional lattice, where the randomness is given by independent, but periodically modulated, inverse-gamma distributed weights. These models interpolate between classical models of disordered systems exhibiting KPZ universality and integrable probabilistic systems with deep connections to random matrix theory, special functions, and representation theory. Periodicity in the environmental parameters introduces new algebraic invariances and combinatorial structures, combining techniques from integrable probability, representation theory, and algebraic combinatorics.

1. Definition and Structure

A periodic inverse-gamma polymer is defined on a lattice (typically $\mathbb{Z} \times \mathbb{Z}$) where each site $(i,j)$ is assigned a random weight $W_{(i,j)}$, drawn independently according to a log-inverse-gamma distribution whose shape parameter is a periodic function of $i$ and $j$. Formally, fix a period $N \in \mathbb{N}$ and two sequences $(a_i)_{i \in \mathbb{Z}}$, $(b_j)_{j \in \mathbb{Z}}$ such that $a_i + b_j > 0$ for all $i,j$. Then,

$$W_{(i,j)} \sim \mathrm{Log}\text{-}\mathrm{Inv}\text{-}\Gamma(a_i+b_j, \lambda)$$

with $a_i$ and $b_j$ extended periodically modulo $N$; $\lambda > 0$ is the scale parameter.

The partition function between two points $(i,j)$ and $(n,m)$ is

$$Z^{W}(n,m \mid i,j) = \sum_{\pi \in \Pi_{(i,j),(n,m)}} \prod_{(r,\ell) \in \pi} \exp(W_{(r,\ell)})$$

where $\Pi_{(i,j),(n,m)}$ denotes directed up-right paths. Multipath and half-flat versions are defined analogously, subject to periodicity constraints.

This periodic inhomogeneity enriches the model by incorporating algebraic symmetries absent in the homogeneous (i.i.d.) setting.

2. Discrete Periodic Pitman Transform and Invariance

A distinguishing analytic tool for the periodic inverse-gamma polymer is the "discrete periodic Pitman transform" (Engel et al., 7 Aug 2025), an operator acting on the (bi-infinite) sequence of periodic weight vectors that preserves single- and multi-path partition functions. Precisely, if $W = (W_k)_{k \in \mathbb{Z}}$ is the configuration (with $W_k$ an $N$-periodic vector), the transform involves two maps $T$ and $D$ on pairs of such vectors: $$T_i(W_2, W_1) = W_{1,i} + \log\left(\frac{\sum_{j \in \mathbb{Z}}\exp([W_2 - W_1]_{[i,j]})}{\sum_{j \in \mathbb{Z}}\exp([W_2 - W_1]_{[i+1,j]})}\right)$$ with analogous formula for $D$. The operator $P_k$ acts by replacing $(W_k, W_{k+1})$ by $(T(W_{k+1}, W_k), D(W_{k+1}, W_k))$.

Main algebraic facts:

• The multi-path partition function is invariant under any product of these transforms, subject to the non-overlap condition.
• The $P_k$ satisfy braid relations $P_k P_{k+1} P_k = P_{k+1} P_k P_{k+1}$ and are involutions.
• At zero temperature, these sum-log transformations become max-plus operations, preserving a corresponding last-passage percolation structure.

These algebraic invariances reflect deep connections with the geometric RSK correspondence and symmetric group actions.

3. Multi-Path Partition Functions and Permutation Invariance

Periodic inverse-gamma polymers admit multi-path partition functions that count weighted collections of non-intersecting paths. Let $U, V$ be $m$-tuples of starting and ending points, the $m$-path partition function is

$$Z^{W}(V | U) = \sum_{\pi \in \Pi_{U,V}} \prod_{(r,\ell) \in \pi} \exp(W_{(r,\ell)})$$

where $\Pi_{U,V}$ consists of $m$ non-intersecting up-right lattice paths.

The discrete periodic Pitman transforms induce an invariance of $Z^{W}(V|U)$ under any reordering of the parameters $(a_i),(b_j)$, effected via a suitable product of $P_k$ operators (Engel et al., 7 Aug 2025). This invariance extends the classical full-line Pitman invariance and is crucial for classifying jointly invariant measures (see below).

4. Burke Properties and Integrability

Periodic inverse-gamma polymers preserve a generalized version of the Burke property (Engel et al., 7 Aug 2025), which in queuing and polymer models refers to a product-form invariance under queueing dynamics or, in this context, the invariance in distribution under the periodic Pitman transform: $$(X_1, X_2) \stackrel{d}{=} (T(X_1, X_2), D(X_1, X_2))$$ where $X_1, X_2$ are periodic weight vectors drawn from the log-inverse-gamma distribution with periodic parameters.

This property ensures that entering the system with given ratios or weights results in output (after the "queueing" step) with the same joint distribution, thus enabling exact computations for partition function statistics and enabling classification of invariant measures.

Under certain regularity and non-vanishing Jacobian assumptions, the class of integrable periodic polymers is exhausted (up to trivial modifications such as scaling and reflection) by such beta-gamma models, including the periodic inverse-gamma case (Chaumont et al., 2017).

5. Explicit Formulas, Jointly Invariant Measures, and Combinatorics

Owing to these invariances, one can explicitly describe the joint law of pairs of weights under the discrete periodic Pitman dynamics. For instance, for the pair $(X_1, D(X_1, X_2))$—with $X_1$ and $X_2$ uniformly distributed over cyclically constrained integer compositions (of specified parameters)—there is an explicit formula for their joint distribution as a function counting alternating segments in the cyclic comparison between $X_1$ and $X_2$ (Engel et al., 7 Aug 2025). For $N=2$ and $N=3$, these formulas involve partitioning the cycle into equality and inequality segments and evaluating products over integer intervals determined by these segments.

Such combinatorial descriptions not only facilitate theoretical analysis but also have implications for algorithmic sampling and for the explicit construction of invariant measures for associated Markov processes.

6. Connections to KPZ Universality, Stationary Solutions, and Scaling Limits

Periodic inverse-gamma polymers retain the scaling exponents characteristic of the Kardar–Parisi–Zhang universality class:

• Fluctuations in the free energy (log-partition function) scale with exponent $1/3$ (variance $\sim N^{2/3}$).
• Transversal fluctuations of the polymer path scale with exponent $2/3$.
• In the long-polymer limit, the centered and scaled free energy converges to Tracy–Widom distributions appropriate for the geometry (GUE/GOE), as made precise in rigorous asymptotic results (1406.59631711.08432Rassoul-Agha et al., 2023).

Periodic analogs of stationary measures and the one force–one solution property have been constructed in both discrete and continuum KPZ models: the measures are joint transforms of independent Brownian bridges (or their semi-discrete analogs) that are invariant under the combined stochastic dynamics (Corwin et al., 5 Sep 2024). In discrete models, these are realized via combinatorial and algebraic objects associated with the periodic RSK and periodic Pitman transforms.

The invariance properties and combinatorial structure persist in the zero-temperature (last-passage percolation) limit, realized by replacing softmax/log operations with max-plus analogs.

7. Significance and Outlook

The periodic inverse-gamma polymer stands at the intersection of integrability, random matrix theory, and stochastic growth. Periodicity introduces braid group symmetries (reflected in the braid relations of the Pitman transform), multi-path invariant structures, and combinatorial complexity that is tractable via algebraic techniques.

The recent development of explicit jointly invariant measures for periodic KPZ-type systems, together with braid invariance and combinatorial formulas, suggests avenues for analyzing more general inhomogeneous, multi-parameter or interacting polymer systems, and for understanding transport and coalescence phenomena in periodic media. At a technical level, these models provide a proving ground for the intersection of exactly solvable models, advanced combinatorics, and probabilistic scaling limits.

The current trajectory points toward a more systematic classification of periodic (and quasi-periodic) integrable models, extensions to other types of symmetries and boundary conditions, and deeper links with representation theory—especially as the focus shifts to full-space and semi-discrete scaling limits, the construction of the directed landscape, and the KPZ fixed point in periodic environments.