Raise and Peel Models of fluctuating interfaces and combinatorics of Pascal's hexagon

TL;DR

This paper extends the raise and peel model of fluctuating interfaces by adding boundary sources, explores their spectra via conformal field theory, and uncovers combinatorial properties linked to Pascal's hexagon and Hirota's equation.

Contribution

It introduces new boundary-extended models, connects their stationary state probabilities to Pascal's hexagon, and reveals their relation to integrable difference equations.

Findings

Spectra described by conformal field theory in the thermodynamic limit.

Stationary state probabilities relate to Pascal's hexagon and combinatorial structures.

Pascal's hexagon solutions connect to Hirota's difference equation.

Abstract

The raise and peel model of a one-dimensional fluctuating interface (model A) is extended by considering one source (model B) or two sources (model C) at the boundaries. The Hamiltonians describing the three processes have, in the thermodynamic limit, spectra given by conformal field theory. The probability of the different configurations in the stationary states of the three models are not only related but have interesting combinatorial properties. We show that by extending Pascal's triangle (which gives solutions to linear relations in terms of integer numbers), to an hexagon, one obtains integer solutions of bilinear relations. These solutions give not only the weights of the various configurations in the three models but also give an insight to the connections between the probability distributions in the stationary states of the three models. Interestingly enough, Pascal's hexagon also gives solutions to a Hirota's difference equation.