Observables of coloured stochastic vertex models and their polymer limits

Coloured stochastic vertex models and their spectral theory

This work is dedicated to $\mathfrak{sl}_{n+1}$-related integrable stochastic vertex models; we call such models coloured. We prove several results about these models, which include the following: (1) We construct the basis of (rational) eigenfunctions of the coloured transfer-matrices as partition functions of our lattice models with certain boundary conditions. Similarly, we construct a dual basis and prove the corresponding orthogonality relations and Plancherel formulae; (2) We derive a variety of combinatorial properties of those eigenfunctions, such as branching rules, exchange relations…

On a coloured tree with non i.i.d. random labels

We obtain new results for the probabilistic model introduced in Menshikov et al (2007) and Volkov (2006) which involves a $d$-ary regular tree. All vertices are coloured in one of $d$ distinct colours so that $d$ children of each vertex all have different colours. Fix $d^2$ strictly positive random variables. For any two connected vertices of the tree assign to the edge between them {\it a label} which has the same distribution as one of these random variables, such that the distribution is determined solely by the colours of its endpoints. {\it A value} of a vertex is defined as a product of …

Hidden diagonal integrability of $q$-Hahn vertex model and Beta polymer model

We study a new integrable probabilistic system, defined in terms of a stochastic colored vertex model on a square lattice. The main distinctive feature of our model is a new family of parameters attached to diagonals rather than to rows or columns, like in other similar models. Because of these new parameters the previously known results about vertex models cannot be directly applied, but nevertheless the integrability remains, and we prove explicit integral expressions for $q$-deformed moments of the (colored) height functions of the model. Following known techniques our model can be interpre…

Shift-invariance for vertex models and polymers

We establish a symmetry in a variety of integrable stochastic systems: Certain multi-point distributions of natural observables are unchanged under a shift of a subset of observation points. The property holds for stochastic vertex models, (1+1)d directed polymers in random media, last passage percolation, the Kardar-Parisi-Zhang equation, and the Airy sheet. In each instance it leads to computations of previously inaccessible joint distributions. The proofs rely on a combination of the Yang-Baxter integrability of the inhomogeneous colored stochastic six-vertex model and Lagrange interpolatio…

二部着色正则图中2步转移概率的不对称性

Asymmetry of 2-step Transit Probabilities in 2-Coloured Regular Graphs

本文研究平衡二着色正则图中红、蓝两类顶点2步停留概率的不对称性，给出一般d-正则图的凸包界，并精确刻画二维环面上的渐近可行域。

Weak-disorder limit for directed polymers on critical hierarchical graphs with vertex disorder

We study models for a directed polymer in a random environment (DPRE) in which the polymer traverses a hierarchical diamond graph and the random environment is defined through random variables attached to the vertices. For these models, we prove a distributional limit theorem for the partition function in a limiting regime wherein the system grows as the coupling of the polymer to the random environment is appropriately attenuated. The sequence of diamond graphs is determined by a choice of a branching number $b\in \{2,3,\ldots\}$ and segmenting number $s\in \{2,3,\ldots\}$, and our focus is o…