Yang-Baxter field for spin Hall-Littlewood symmetric functions

2017/12/13

Alexey Bufetov, Leonid Petrov
Forum of Mathematics Sigma 7 (2019), e39 • arXiv:1712.04584 [math.PR]

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Employing bijectivisation of summation identities, we introduce local stochastic moves based on the Yang-Baxter equation for $U_q(\widehat{\mathfrak{sl}_2})$. Combining these moves leads to a new object which we call the spin Hall-Littlewood Yang-Baxter field - a probability distribution on two-dimensional arrays of particle configurations on the discrete line. We identify joint distributions along down-right paths in the Yang-Baxter field with spin Hall-Littlewood processes, a generalization of Schur processes. We consider various degenerations of the Yang-Baxter field leading to new dynamic versions of the stochastic six vertex model and of the Asymmetric Simple Exclusion Process.

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A poem on the topic

by OpenAI

The stars aloft their burning lamps display,
To light the paths of night away;
And yonder in the Yang-Baxter field,
The spin Hall-Littlewood symmetric yield.

The symmetry of night, so grand and fair,
Is held in harmony by the square;
And in its own eternal way,
Shines forth in every spin Hall day.

The spin Hall-Littlewood symmetric stay,
In constancy through night and day;
In every corner of the sky,
It stands unshaken, bold and high.

The stars above are ever bright,
Their symmetry is ever right;
The spin Hall-Littlewood symmetric yield
Shines in the Yang-Baxter field.

Forward transition probabilities associated with the Yang-Baxter field