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January 2012 Scaling for a one-dimensional directed polymer with boundary conditions
Timo Seppäläinen
Ann. Probab. 40(1): 19-73 (January 2012). DOI: 10.1214/10-AOP617


We study a (1 + 1)-dimensional directed polymer in a random environment on the integer lattice with log-gamma distributed weights. Among directed polymers, this model is special in the same way as the last-passage percolation model with exponential or geometric weights is special among growth models, namely, both permit explicit calculations. With appropriate boundary conditions, the polymer with log-gamma weights satisfies an analogue of Burke’s theorem for queues. Building on this, we prove the conjectured values for the fluctuation exponents of the free energy and the polymer path, in the case where the boundary conditions are present and both endpoints of the polymer path are fixed. For the polymer without boundary conditions and with either fixed or free endpoint, we get the expected upper bounds on the exponents.


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Timo Seppäläinen. "Scaling for a one-dimensional directed polymer with boundary conditions." Ann. Probab. 40 (1) 19 - 73, January 2012.


Published: January 2012
First available in Project Euclid: 3 January 2012

zbMATH: 1254.60098
MathSciNet: MR2917766
Digital Object Identifier: 10.1214/10-AOP617

Primary: 60K35 , 60K37
Secondary: 82B41 , 82D60

Keywords: Burke’s theorem , Directed polymer , Partition function , random environment , Scaling exponent , Superdiffusivity

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 1 • January 2012
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