The endpoint distribution of directed polymers

Bates, Erik; Chatterjee, Sourav

doi:10.48550/arxiv.1612.03443

Cited by 10 publications

(45 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Another feature of the polymer paths in low temperatures is that they localize and concentrate in small regions, see e.g. [6,11,15,32] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

A PDE hierarchy for directed polymers in random environments

Gu,

Henderson

2020

Preprint

View full text Add to dashboard Cite

For a Brownian directed polymer in a Gaussian random environment, where q(t, ⋅) is the quenched endpoint density and Qn(t, x 1 , . . . , xn) = E[q(t, x 1 ) . . . q(t, xn)], we derive a hierarchical PDE system satisfied by {Qn} n≥1 . In d = 1 and for the case of a spacetime white noise environment, we study a nonlocal reaction-diffusion equation that is related to the generator of {µt(dx) = q(t, x)dx} t≥0 , viewed as a Markov process taking values in the space of probability measures, and establish super-diffusive behavior with the spreading occurring on the scale O(t 2 3 ).

show abstract

“…Another feature of the polymer paths in low temperatures is that they localize and concentrate in small regions, see e.g. [6,11,15,32] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

A PDE hierarchy for directed polymers in random environments

Gu,

Henderson

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…This paper continues the advance in this direction by proving that in the localization regime, certain qualitative behaviors of the polymer's endpoint distribution are the same for any reference walk. Namely, the strongest forms of localization known for arbitrary environment and arbitrary dimension, which were only recently proved for the SRW case in [10], are established here for the general case.…”

Section: Introductionmentioning

confidence: 73%

Localization of directed polymers with general reference walk

Bates¹

2018

Electron. J. Probab.

Self Cite

View full text Add to dashboard Cite

Directed polymers in random environment have usually been constructed with a simple random walk on the integer lattice. It has been observed before that several standard results for this model continue to hold for a more general reference walk. Some finer results are known for the so-called long-range directed polymer in which the reference walk lies in the domain of attraction of an α-stable process. In this note, low-temperature localization properties recently proved for the classical case are shown to be true with any reference walk. First, it is proved that the polymer's endpoint distribution is asymptotically purely atomic, thus strengthening the best known result for long-range directed polymers. A second result proving geometric localization along a positive density subsequence is new to the general case. The proofs use a generalization of the approach introduced by the author with S. Chatterjee in a recent manuscript on the quenched endpoint distribution; this generalization allows one to weaken assumptions on the both the walk and the environment. The methods of this paper also give rise to a variational formula for free energy which is analogous to the one obtained in the simple random walk case.2010 Mathematics Subject Classification. 60K37, 82B26, 82B44, 82D60.

show abstract

“…accumulates all its mass for large enough γ (see [CSY03,V07,BC16] for related results on discrete directed polymers, which as in the case of discrete Gaussian free field, is defined pointwise on the lattice without any need of a mollification scheme). Part of the technique in [BM19-II] builds on a fixed point approach from spin glasses used in [BC16] which in this context is reliant upon a particular well-behaved Markovian dynamics that the endpoint distribution Q γ,t defines, the latter being a random element of M 1 (R d ), the space of probability measures on R d . This space is noncompact (under the usual weak topology), which a priori disallows a nice behavior of this Markov chain therein.…”

Section: Concentration Of Gmc Covariance For Large γmentioning

confidence: 99%

Geometry of the Gaussian multiplicative chaos in the Wiener space

Bröker¹,

Mukherjee²

2020

Preprint

View full text Add to dashboard Cite

The theory of Gaussian multiplicative chaos (GMC) is usually defined w.r.t. logcorrelated Gaussian fields in the finite dimensional context and dates back to a seminal work of Kahane [K85] which contains ideas and results that led to a lot of rejuvenated interest in the subject during the last decade. Inspired by ongoing investigations pertinent to the Liouville measure in 2d-Liouville quantum gravity, thick points of the Gaussian free field and volume decay of Liouville balls and scaling exponents, in the present infinite dimensional context we drop all assumptions on log-correlations and consider a centered Gaussian field, driven by space-time white noise and indexed by continuous paths equipped with the law of Brownian paths on R d as reference measure. An exponentiation of this field, followed by suitable renormalization, defines a Gaussian multiplicative chaos on Wiener paths. As long as the coupling constant is tuned sufficiently low (and in this regime it is known that for d ≥ 3, the field attains high values on any path sampled according to the GMC probability measure) and with the white noise field spatially mollified at scale ε > 0, the (renormalized) GMC volume of any microscopic ball of radius ε d/2 in the Wiener space, with its location chosen uniformly therein, decays at least with speed ε d in an almost sure sense as the mollification scheme is turned off. We also show that when the coupling constant is tuned high, the energy landscape of the system freezes and enters the so-called glassy phase as the normalized covariance of the field under the GMC measure concentrates most of its mass in an averaged sense. A key aspect of our proof, while not making any assumptions about log-correlations, builds on Kahane's techniques [K85] in a general set up, combined with geometric properties of the underlying metric space of continuous functions.

show abstract

The endpoint distribution of directed polymers

Cited by 10 publications

References 0 publications

A PDE hierarchy for directed polymers in random environments

A PDE hierarchy for directed polymers in random environments

Localization of directed polymers with general reference walk

Geometry of the Gaussian multiplicative chaos in the Wiener space

Contact Info

Product

Resources

About