Boundary loop models
We present a class of two-dimensional statistical models of loops interacting with the boundary of the system. Each loop touching the boundary receives a weight different from that of bulk loops. This allows for the construction of a continuous family of conformal boundary conditions. The problem has many links with combinatorics, algebra, and representation theory.
As an application we consider vertex colourings of large planar graphs with a boundary. For a fixed number of bulk colours, varying the number of boundary colours leads to multiple phase transitions that we characterise exactly.
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