Cross-Parameter Swapping (CP-SSM)

• Cross-Parameter Swapping (CP-SSM) is a mechanism that uses local swap operators to transpose parameters in integrable stochastic systems such as q-Hahn TASEP and directed beta polymers.
• The mechanism employs explicit Markov coupling and contour-integral moment formulas to maintain ergodicity, duality, and stationarity under parameter exchanges.
• CP-SSM unifies and generalizes local split–mix constructions across models like q-TASEP and the directed beta polymer, offering a versatile framework for studying stochastic dynamics.

The Cross-Parameter Swapping Mechanism (CP-SSM) is a local, explicit operator construction that realizes permutation symmetries of parameter-dependent integrable stochastic systems, particularly those exhibiting multidimensional parameterization such as the $q$-Hahn TASEP, the $q$-TASEP, and the directed beta polymer. In these settings, swapping two adjacent system parameters results—through the action of a swap operator at a single site—in an exact transition to the law of the configuration where those parameters are transposed. The operator manifests as a Markov coupling and is analytically tractable via contour-integral moment formulas, preserving both integrability and essential model characteristics. CP-SSM unifies and generalizes local “split–mix” constructions for a wide class of particle systems and random polymers, providing a fundamental mechanism for studying stationary measures, duality, and ergodicity in exactly solvable models (Petrov, 2019).

1. Parameter-Symmetric Systems and Swap Operators

A parameter-symmetric stochastic particle system consists of configurations $\mathbf x^{\boldsymbol\nu}(t)$ on the space $\mathrm{Conf}_{\rm fin}(\mathbb{Z})$, with parameters $\boldsymbol\nu = (\nu_1, \nu_2, \dots)$. A system is parameter-symmetric if, for each adjacent transposition $s_n = (n, n+1)$, swapping $\nu_n \leftrightarrow \nu_{n+1}$ changes only the distribution of $x_n(t)$, leaving the joint law of all other particles invariant.

Formally, there exists a local stochastic kernel—called the swap operator—denoted $p_n(x_n'|x_{n-1}, x_n, x_{n+1})$, such that substituting $x_n$ with $x_n'$ (conditioned on its neighbors) yields a new configuration distributed as if the corresponding parameters were transposed. The locality of $p_n$ is critical: the operator depends only on adjacent gaps and acts trivially on the rest of the system (Petrov, 2019).

2. Realization in $q$-Hahn TASEP and Limiting Models

In discrete-time $q$-Hahn TASEP, each particle $x_i$ with $0<\nu_i<1$ and global parameter $\gamma\geq 1$ jumps right by $j$ with probability

$$\varphi_{q, \gamma\nu_i, \nu_i}(j \mid x_{i-1} - x_i - 1),$$

where $\varphi_{q, \mu, \nu}$ is the $q$-deformed beta-binomial distribution. Integrability yields contour-integral moment representations for such systems.

The CP-SSM swap operator at site $n$, under $\nu_{n+1}<\nu_n$, is given by

$$p^{\rm qH}_n(x_n' \mid x_{n-1}, x_n, x_{n+1}) = \varphi_{q,\, \nu_{n+1}/\nu_n,\, \nu_{n+1}}(x_n' - x_{n+1} - 1 \mid x_n - x_{n+1} - 1),$$

supported on $x_{n+1} + 1 \leq x_n' \leq x_n$. Substitution of this kernel into the moment formula confirms that $q^{x_n+n}$ is updated in law as if $\nu_n$ and $\nu_{n+1}$ are exchanged (Petrov, 2019).

Continuous-time and degenerate cases include the $q$-TASEP ($\nu=0$ limit) and the backwards Hammersley process ($q\to0$). CP-SSM thereby encompasses a broad class of systems under a unified local operation.

3. Stationary Dynamics and Backward Evolution

Repeated application of swap operators across sites and in continuous-time limits constructs backward evolutions, such as the backward $q$-Hahn process and the backward Hammersley process. Sequentially applying swap operators $(p_1, p_2, \dots)$ in a system with parameters $\nu_n = \nu r^{n-1}$ and passing to a Poisson scaling limit yields an inhomogeneous, continuous-time Markov process where values jump left across available sites at rates described explicitly by CP-SSM-induced kernels.

In the $q$-TASEP ($\nu=0$), the forward generator $\mathsf T$ and the backward generator $\mathsf B$ combine to form a one-parameter stationary Markov generator,

$$\mathsf Q^{(T)} = \mathsf T + \frac{1}{T} \mathsf B,$$

where $T$ is fixed and $\mathsf Q^{(T)}$ governs ergodic dynamics preserving the $q$-TASEP law at time $T$ (Petrov, 2019).

4. Duality, Moments, and Ergodicity

CP-SSM produces dual Markov processes essential for rigorous study of moment formulas and convergence. The generator $\mathsf Q^{(T)}$ for the particle system is dual (under $$H(\mathbf x, \mathbf n) = \prod_i q^{x_{n_i} + n_i}$$) to a combination of the stochastic $q$-Boson process and a novel transient process $\breve{\mathsf B}$. The survival probabilities in the dual encode the asymptotic $q$-moments of the $q$-TASEP, uniquely determined by harmonic equations and boundary conditions matching those of the integrable system.

This duality establishes ergodicity: as the Markov process evolves, the system converges in law to the $q$-TASEP at time $T$, with the limiting moment structure directly computable from the dual survival probabilities (Petrov, 2019).

5. CP-SSM in the Directed Beta Polymer

In the directed beta polymer on the strict-weak lattice, the partition function $Z(t, n)$ evolves via random $\mathrm{Beta}(\nu_n-\gamma, \gamma)$-distributed weights. The CP-SSM swap operator at site $n$ is realized by introducing an independent beta variable $$\tilde{B} \sim \mathrm{Beta}(\nu_{n+1}-\nu_n,\, \nu_n)$$ and updating

$$Z(t, n) \longmapsto \tilde{Z}(t, n) = \tilde{B} Z(t, n+1) + (1 - \tilde{B}) Z(t, n).$$

This linear mixing ensures that swapping $\nu_n \leftrightarrow \nu_{n+1}$ leaves the joint law invariant up to the transposition, as verified by matching moment integral and boundary condition arguments with those for $q$-Hahn TASEP (Petrov, 2019).

A lattice-level interpretation is available: the swap corresponds to the addition of a new vertex and edge weights defined by $\tilde{B}$, re-routing the partition function and effecting the parameter exchange in distribution.

6. Limiting Cases and Structural Unity

CP-SSM unifies a variety of stochastic models under a local swap paradigm:

• In the $q\to0$ ($\nu\to0$ in $q$-Hahn) limit, it recovers standard TASEP with particle-dependent speeds and the backward Hammersley process.
• The $q\to1$, $\nu,\gamma\to1$ regime produces the (log-)gamma polymer, with beta-mixing corresponding to parameter shifts in the gamma distribution.
• The same moment-integral and boundary-condition verifications extend across these limits, indicating the structural role of CP-SSM as a universal “split–mix” mechanism throughout the hierarchy of solvable random polymer and exclusion models (Petrov, 2019).

7. Summary and Significance

The Cross-Parameter Swapping Mechanism provides a rigorous, local, and integrable means to transpose parameters in stochastic particle systems and directed polymers, realized via explicit Markov swap operators that act only at a single site. The essential method is to connect these local moves, via known contour-integral formulas and key combinatorial symmetries, to global parameter permutations in the model’s law. By iterating swap moves one builds backward evolution operators, which combine with forward dynamics to yield stationary Markov processes whose duals capture the full moment information and guarantee ergodicity to the system’s time-marginal law. This construction encapsulates and generalizes previous approaches to parameter permutation symmetry, providing a versatile analytic framework for future investigations of integrable stochastic processes (Petrov, 2019).