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Limit shapes for inhomogeneous corner growth models with exponential and geometric weights. (English) Zbl 1338.60229

Summary: We generalize the exactly solvable corner growth models by choosing the rate of the exponential distribution \(a_i+b_j\) and the parameter of the geometric distribution \(a_i b_j\) at site \((i, j)\), where \((a_i)_{i \geq 1}\) and \((b_j)_{j \geq 1}\) are jointly ergodic random sequences. We identify the shape function in terms of a simple variational problem, which can be solved explicitly in some special cases.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60K37 Processes in random environments