Abstract
We show that nontrivial bi-infinite polymer Gibbs measures do not exist in typical environments in the inverse-gamma (or log-gamma) directed polymer model on the planar square lattice. The precise technical result is that, except for measures supported on straight-line paths, such Gibbs measures do not exist in almost every environment when the weights are independent and identically distributed inverse-gamma random variables. The proof proceeds by showing that when two endpoints of a point-to-point polymer distribution are taken to infinity in opposite directions but not parallel to lattice directions, the midpoint of the polymer path escapes. The proof is based on couplings, planar comparison arguments, and a recently discovered joint distribution of Busemann functions.
Funding Statement
O. Busani was supported by EPSRC’s EP/R021449/1 Standard Grant. T. Seppäläinen was partially supported by National Science Foundation grant DMS-1854619 and by the Wisconsin Alumni Research Foundation.
Citation
Ofer Busani. Timo Seppäläinen. "Non-existence of bi-infinite polymers." Electron. J. Probab. 27 1 - 40, 2022. https://doi.org/10.1214/21-EJP731
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