KPZ-type fluctuation exponents for interacting diffusions in equilibrium

Abstract

We study the fluctuations in equilibrium of a class of Brownian motions interacting through a potential. For a certain choice of exponential potential, the distribution of the system coincides with differences of free energies of the stationary semi-discrete or O'Connell-Yor polymer.We show that for Gaussian potentials, the fluctuations are of order N 1 4 when the time and system size coincide, whereas for a class of more general convex potentials V the fluctuations are of order at most N 1 3 . In the O'Connell-Yor case, we recover the known upper bounds for the fluctuation exponents using a dynamical approach, without reference to the polymer partition function interpretation.

Another look at the Balázs-Quastel-Seppäläinen theorem

“…[27,28,14,1,9,6,7,8,12,4,29] for relevant results of proving the scaling exponents, deriving the Tracy-Widom type fluctuations etc. Our proof of the upper bound is inspired by the approach used to study the O'Connell-Yor polymer in [28], which was further explored in [20] (see also the recent study on the interacting diffusions [17] using a similar strategy).…”

Upper tail bounds for stationary KPZ models

“…Through an analogous argument, we also obtain the same result for the upper tail of a height function defined in terms of a model of Brownian motions interacting through a potential, at equilibrium. We had previously studied the fluctuations of the latter model at the level of the second moment with C. Noack in [18]. For a specific choice of interaction potential, the height function in this model coincides in distribution with the log-partition function of the O'Connell-Yor semi-discrete polymer, but in general the model is not expected to be integrable.…”

Coupling derivation of optimal-order central moment bounds in exponential last-passage percolation

“…One of the earliest examples of this is the observation due to [105] that the exponential LPP process fluctuates with exponent at most 1{3 above its limit shape as a consequence of the LPP superadditivity and the structure of the right-tail large deviation rate function, which can be computed exactly from the invariant measures in the exponential case. The existence of a one-parameter family of invariant measures has been shown for various (supposedly) KPZ-class models including some general versions of the planar LPP [9,10,27,61,62,88] and directed polymers [11,66,67], as well as broad classes of interacting particles [1,12,15,24,34,53,54,55,82,84,85,89]. In a variety of examples [13,14,16,17,18,26,31,33,78,82,91,92,93,108,110] with product-form invariant measures, it has been possible to adapt and build on the probabilistic coupling approach of [26] to derive KPZ-type fluctuation bounds.…”

“…The existence of a one-parameter family of invariant measures has been shown for various (supposedly) KPZ-class models including some general versions of the planar LPP [9,10,27,61,62,88] and directed polymers [11,66,67], as well as broad classes of interacting particles [1,12,15,24,34,53,54,55,82,84,85,89]. In a variety of examples [13,14,16,17,18,26,31,33,78,82,91,92,93,108,110] with product-form invariant measures, it has been possible to adapt and build on the probabilistic coupling approach of [26] to derive KPZ-type fluctuation bounds. A few of these results cover nonintegrable models [14,15,82] at equilibrium (initialized with an invariant measure).…”

“…In a variety of examples [13,14,16,17,18,26,31,33,78,82,91,92,93,108,110] with product-form invariant measures, it has been possible to adapt and build on the probabilistic coupling approach of [26] to derive KPZ-type fluctuation bounds. A few of these results cover nonintegrable models [14,15,82] at equilibrium (initialized with an invariant measure). Although the invariant measures associated with directed percolation and polymers are not of product-form in general, sufficient understanding of these measures can plausibly lead to the KPZ universality at least at the level of exponents.…”