TY - JOUR

T1 - Geometrical optics of constrained Brownian excursion

T2 - From the KPZ scaling to dynamical phase transitions

AU - Smith, Naftali R.

AU - Meerson, Baruch

N1 - Funding Information:
We acknowledge useful discussions with Tal Agranov, Mark Dykman, Patrik Ferrari and Senya Shlosman. This research was supported by the Israel Science Foundation (grant No. 807/16). NRS was supported by the Clore Foundation.
Publisher Copyright:
© 2019 IOP Publishing Ltd and SISSA Medialab srl.

PY - 2019/2/25

Y1 - 2019/2/25

N2 - We study a Brownian excursion on the time interval |t| T , conditioned to stay above a moving wall x0 (t) such that x0 (-T) = x0 (T) = 0, and x0 (|t| T) 0. For a whole class of moving walls, typical fluctuations of the conditioned Brownian excursion are described by the Ferrari-Spohn (FS) distribution and exhibit the Kardar-Parisi-Zhang (KPZ) dynamic scaling exponents 1/3 and 2/3. Here we use the optimal fluctuation method (OFM) to study atypical fluctuations, which turn out to be quite different. The OFM provides their simple description in terms of optimal paths, or rays, of the Brownian motion. We predict two singularities of the large deviation function, which can be interpreted as dynamical phase transitions, and they are typically of third order. Transitions of a fractional order can also appear depending on the behavior of x0 (t) in a close vicinity of t = T. Although the OFM does not describe typical fluctuations, it faithfully reproduces the near tail of the FS distribution and therefore captures the KPZ scaling. If the wall function x0 (t) is not parabolic near its maximum, typical fluctuations (which we probe in the near tail) exhibit a more general scaling behavior with a continuous oneparameter family of scaling exponents.

AB - We study a Brownian excursion on the time interval |t| T , conditioned to stay above a moving wall x0 (t) such that x0 (-T) = x0 (T) = 0, and x0 (|t| T) 0. For a whole class of moving walls, typical fluctuations of the conditioned Brownian excursion are described by the Ferrari-Spohn (FS) distribution and exhibit the Kardar-Parisi-Zhang (KPZ) dynamic scaling exponents 1/3 and 2/3. Here we use the optimal fluctuation method (OFM) to study atypical fluctuations, which turn out to be quite different. The OFM provides their simple description in terms of optimal paths, or rays, of the Brownian motion. We predict two singularities of the large deviation function, which can be interpreted as dynamical phase transitions, and they are typically of third order. Transitions of a fractional order can also appear depending on the behavior of x0 (t) in a close vicinity of t = T. Although the OFM does not describe typical fluctuations, it faithfully reproduces the near tail of the FS distribution and therefore captures the KPZ scaling. If the wall function x0 (t) is not parabolic near its maximum, typical fluctuations (which we probe in the near tail) exhibit a more general scaling behavior with a continuous oneparameter family of scaling exponents.

KW - Brownian motion

KW - dynamical processes

KW - large deviations in non-equilibrium systems

UR - http://www.scopus.com/inward/record.url?scp=85062550940&partnerID=8YFLogxK

U2 - 10.1088/1742-5468/ab00e8

DO - 10.1088/1742-5468/ab00e8

M3 - Article

AN - SCOPUS:85062550940

VL - 2019

JO - Journal of Statistical Mechanics: Theory and Experiment

JF - Journal of Statistical Mechanics: Theory and Experiment

SN - 1742-5468

IS - 2

M1 - 023205

ER -