TY - JOUR

T1 - Nonexistence of Bigeodesics in Planar Exponential Last Passage Percolation

AU - Basu, Riddhipratim

AU - Hoffman, Christopher

AU - Sly, Allan

N1 - Funding Information:
We thank Manjunath Krishnapur for pointing us to the reference [], Daniel Ahlberg for useful discussions at the early stage of the project, and Yiping Hu for a careful reading of the paper. We also thank two anonymous referees for many helpful comments and suggestions. RB is partially supported by a Ramanujan Fellowship (SB/S2/RJN-097/2017) from the Science and Engineering Research Board, an ICTS-Simons Junior Faculty Fellowship, DAE project no. RTI4001 via ICTS and Infosys Foundation via the Infosys-Chandrasekharan Virtual Centre for Random Geometry of TIFR. CH is supported by a Simons fellowship and NSF Grants DMS-1712701 and DMS-1954059. AS is supported by NSF grant DMS-1855527, a Simons Investigator grant and a MacArthur Fellowship. Part of this research was performed during a visit of RB to the Princeton Mathematics department whose hospitality he gratefully acknowledges.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2022/1

Y1 - 2022/1

N2 - Bi-infinite geodesics are fundamental objects of interest in planar first passage percolation. A longstanding conjecture states that under mild conditions there are almost surely no bigeodesics; however, the result has not been proved in any case. For the exactly solvable model of directed last passage percolation on Z2 with i.i.d. exponential passage times, we study the corresponding question and show that almost surely the only bigeodesics are the trivial ones, i.e., the horizontal and vertical lines. The proof makes use of estimates for last passage time available from the integrable probability literature to study coalescence structure of finite geodesics, thereby making rigorous a heuristic argument due to Newman (Auffinger et al., 50 Years of First-passage Percolation, American Mathematical Soc., 2017).

AB - Bi-infinite geodesics are fundamental objects of interest in planar first passage percolation. A longstanding conjecture states that under mild conditions there are almost surely no bigeodesics; however, the result has not been proved in any case. For the exactly solvable model of directed last passage percolation on Z2 with i.i.d. exponential passage times, we study the corresponding question and show that almost surely the only bigeodesics are the trivial ones, i.e., the horizontal and vertical lines. The proof makes use of estimates for last passage time available from the integrable probability literature to study coalescence structure of finite geodesics, thereby making rigorous a heuristic argument due to Newman (Auffinger et al., 50 Years of First-passage Percolation, American Mathematical Soc., 2017).

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U2 - 10.1007/s00220-021-04246-0

DO - 10.1007/s00220-021-04246-0

M3 - Article

AN - SCOPUS:85119831034

SN - 0010-3616

VL - 389

JO - Communications In Mathematical Physics

JF - Communications In Mathematical Physics

IS - 1

ER -