On the limiting law of line ensembles of Brownian polymers with geometric area tilts
Amir Dembo, Eyal Lubetzky, Ofer Zeitouni
备注：
Revision adds treatment of non-integer $T$, allows for boundary measures different than Lebesgue, and addresses referees comments. To appear in Annals Inst. H. Poincare
We study the line ensembles of non-crossing Brownian bridges above a hard wall, each tilted by the area of the region below it with geometrically growing pre-factors. This model, which mimics the level lines of the $(2+1)$D SOS model above a hard wall, was studied in two works from 2019 by Caputo, Ioffe and Wachtel. In those works, the tightness of the law of the top $k$ paths, for any fixed $k$, was established under either zero or free boundary conditions, which in the former setting implied the existence of a limit via a monotonicity argument. Here we address the open problem of a limit under free boundary conditions: we prove that as the interval length, followed by the number of paths, go to $\infty$, the top $k$ paths converge to the same limit as in the free boundary case, as conjectured by Caputo, Ioffe and Wachtel.
In this paper, we introduce a model of Brownian polymer in a continuous random environment. The asymptotic behavior of the partition function associated to this polymer measure is studied, and we are able to separate a weak and strong disorder regime under some reasonable assumptions on the spatial covariance of the environment. Some further developments, concerning some concentration inequalities for the partition function, are given for the weak disorder regime.
We study a continuum model of directed polymer in random environment. The law of the polymer is defined as the Brownian motion conditioned to survive among space-time Poissonian disasters. This model is well-studied in the positive temperature regime. However, at zero-temperature, even the existence of the free energy has not been proved. In this article, we prove that the free energy exists and is continuous at zero-temperature.
This paper provides information about the asymptotic behavior of a one-dimensional Brownian polymer in random medium represented by a space-time Gaussian field W assumed to be white noise in time and function-valued in space. According to the behavior of the spatial covariance W, we give a lower bound on the power growth (wandering exponent) of the polymer when the time parameter goes to infinity: the polymer is proved to be superdiffusive, with a wandering exponent exceeding any $α<3/5$.
This paper provides information about the asymptotic behavior of a one-dimensional Brownian polymer in random medium represented by a Gaussian field $W$ on ${\mathbb{R}}_+\times{\mathbb{R}}$ which is white noise in time and function-valued in space. According to the behavior of the spatial covariance of $W$, we give a lower bound on the power growth (wandering exponent) of the polymer when the time parameter goes to infinity: the polymer is proved to be superdiffusive, with a wandering exponent exceeding any $α<3/5$.
We consider a model of the shape of a growing polymer introduced by Durrett and Rogers (Probab. Theory Related Fields 92 (1992) 337--349). We prove their conjecture about the asymptotic behavior of the underlying continuous process $X_t$ (corresponding to the location of the end of the polymer at time $t$) for a particular type of repelling interaction function without compact support.
We estimate the mean first time, called the mean rotation time (MRT), for a planar random polymer to wind around a point. This polymer is modeled as a collection of n rods, each of them being parameterized by a Brownian angle. We are led to study the sum of i.i.d. imaginary exponentials with one dimensional Brownian motions as arguments. We find that the free end of the polymer satisfies a novel stochastic equation with a nonlinear time function. Finally, we obtain an asymptotic formula for the MRT, whose leading order term depends on the square root of n and, interestingly, depends weakly on …
We study the large-time behavior of the charged-polymer Hamiltonian $H_n$ of Kantor and Kardar [Bernoulli case] and Derrida, Griffiths, and Higgs [Gaussian case], using strong approximations to Brownian motion. Our results imply, among other things, that in one dimension the process $\{H_{[nt]}\}_{0\le t\le 1}$ behaves like a Brownian motion, time-changed by the intersection local-time process of an independent Brownian motion. Chung-type LILs are also discussed.
We consider tightness for families of non-colliding Brownian bridges above a hard wall, which are subject to geometrically growing self-potentials of tilted area type. The model is introduced in order to mimic level lines of $2+1$ discrete Solid-On-Solid random interfaces above a hard wall.
We consider two models of random diffusion in random environment in two dimensions. The first example is the self-repelling Brownian polymer, this describes a diffusion pushed by the negative gradient of its own occupation time measure (local time). The second example is a diffusion in a fixed random environment given by the curl of massless Gaussian free field. In both cases we show that the process is superdiffusive: the variance grows faster than linearly with time. We give lower and upper bounds of the order of t log log t, respectively, t log t. We also present computations for an anisotr…
We prove tightness and limiting Brownian-Gibbs description for line ensembles of non-colliding Brownian bridges above a hard wall, which are subject to geometrically growing self-potentials of tilted area type. Statistical properties of the resulting ensemble are very different from that of non-colliding Brownian bridges without self-potentials. The model itself was introduced in order to mimic level lines of (2+1)-dimensional discrete Solid-On-Solid random interfaces above a hard wall.
We prove a formula conjectured in O'Connell and Yor (2001) for the free energy density of a directed polymer in a Brownian environment in 1+1 dimensions.
This paper proves an equality in law between the invariant measure of a reflected system of Brownian motions and a vector of point-to-line last passage percolation times in a discrete random environment. A consequence describes the distribution of the all-time supremum of Dyson Brownian motion with drift. A finite temperature version relates the point-to-line partition functions of two directed polymers, with an inverse-gamma and a Brownian environment, and generalises Dufresne's identity. Our proof introduces an interacting system of Brownian motions with an invariant measure given by a field…
We consider the Brownian directed polymer in Poissonian environment in dimension 1+1, under the so-called intermediate disorder regime, which is a crossover regime between the strong and weak disorder regions. We show that, under a diffusive scaling involving different parameters of the system, the renormalized point-to-point partition function of the polymer converges in law to the solution of the stochastic heat equation with Gaussian multiplicative noise. The Poissonian environment provides a natural setting and strong tools, such as the Wiener-Ito chaos expansion, which, applied to the par…
We study the scaling exponents of a 1+1-dimensional directed polymer in a Brownian random environment introduced by O'Connell and Yor. For a version of the model with boundary conditions that are stationary in a space-time sense we identify the exact values of the exponents. For the version without the boundary conditions we get the conjectured upper bounds on the exponents.
The model of Brownian Percolation has been introduced as an approximation of discrete last-passage percolation models close to the axis. It allowed to compute some explicit limits and prove fluctuation theorems for these, based on the relations between the Brownian percolation and random matrices. Here, we present two approaches that allow to treat discrete asymmetric models of directed polymers. In both cases, the behaviour is universal, meaning that the results do not depend on the precise law of the environment as long as it satisfies some natural moment assumptions. First, we establish an …
This paper is concerned with two related types of directed polymers in a random medium. The first one is a d-dimensional Brownian motion living in a random environment which is Brownian in time and homogeneous in space. The second is a continuous-time random walk on the lattice Z^d, in a random environment with similar properties as in continuous space. The case of a space-time white noise environment can be acheived in this second setting. By means of some Gaussian tools, we estimate the free energy of these models at low temperature, and give some further information on the strong disorder r…
In last passage percolation models lying in the KPZ universality class, the energy of long energy-maximizing paths may be studied as a function of the paths' pair of endpoint locations. Scaled coordinates may be introduced, so that these maximizing paths, or polymers, now cross unit distances with unit-order fluctuations, and have scaled energy, or weight, of unit order. In this article, we consider Brownian last passage percolation in these scaled coordinates. In the narrow wedge case, when one endpoint of such polymers is fixed, say at $(0,0) \in \mathbb{R}^2$, and the other is varied horizo…
In last passage percolation models lying in the KPZ universality class, long maximizing paths have a typical deviation from the linear interpolation of their endpoints governed by the two-thirds power of the interpolating distance. This two-thirds power dictates a choice of scaled coordinates, in which these maximizers, now called polymers, cross unit distances with unit-order fluctuations. In this article, we consider Brownian last passage percolation in these scaled coordinates, and prove that the probability of the presence of $k$ disjoint polymers crossing a unit-order region while beginni…
We study a quenched charged-polymer model, introduced by Garel and Orland in 1988, that reproduces the folding/unfolding transition of biopolymers. We prove that, below the critical inverse temperature, the polymer is delocalized in the sense that: (1) The rescaled trajectory of the polymer converges to the Brownian path; and (2) The partition function remains bounded. At the critical inverse temperature, we show that the maximum time spent at points jumps discontinuously from 0 to a positive fraction of the number of monomers, in the limit as the number of monomers tends to infinity. Finally,…
We study the asymptotic behavior of a self-interacting one-dimensional Brownian polymer first introduced by Durrett and Rogers [Probab. Theory Related Fields 92 (1992) 337--349]. The polymer describes a stochastic process with a drift which is a certain average of its local time. We show that a smeared out version of the local time function as viewed from the actual position of the process is a Markov process in a suitably chosen function space, and that this process has a Gaussian stationary measure. As a first consequence, this enables us to partially prove a conjecture about the law of larg…
We consider the Anderson polymer partition function $$ u(t):=\mathbb{E}^X\Bigl[e^{\int_0^t \mathrm{d}B^{X(s)}_s}\Bigr]\,, $$ where $\{B^{x}_t\,;\, t\geq0\}_{x\in\mathbb{Z}^d}$ is a family of independent fractional Brownian motions all with Hurst parameter $H\in(0,1)$, and $\{X(t)\}_{t\in \mathbb{R}^{\geq 0}}$ is a continuous-time simple symmetric random walk on $\mathbb{Z}^d$ with jump rate $κ$ and started from the origin. $\mathbb{E}^X$ is the expectation with respect to this random walk. We prove that when $H\leq 1/2$, the function $u(t)$ almost surely grows asymptotically like $e^{l t}$, wh…
In last passage percolation models lying in the KPZ universality class, the energy of long energy-maximizing paths may be studied as a function of the paths' pair of endpoint locations. Scaled coordinates may be introduced, so that these maximizing paths, or polymers, now cross unit distances with unit-order fluctuations, and have scaled energy, or weight, of unit order. In this article, we consider Brownian last passage percolation in these scaled coordinates. In the narrow wedge case, one endpoint of such polymers is fixed, say at $(0,0) \in \mathbb{R}^2$, and the other is varied horizontall…
In the context of the coloured stochastic vertex model in a quadrant, we identify a family of observables whose averages are given by explicit contour integrals. The observables are certain linear combinations of $q$-moments of the coloured height functions of the model. In a polymer limit, this yields integral representations for moments of partition functions of strict-weak, semi-discrete Brownian, and continuum Brownian polymers with varying beginning and ending points of the polymers.
We prove that given any fixed asymptotic velocity, the finite length O'Connell-Yor polymer has an infinite length limit satisfying the law of large numbers with this velocity. By a Markovian property of the quenched polymer this reduces to showing the existence of Busemann functions: almost sure limits of ratios of random point-to-point partition functions. The key ingredients are the Burke property of the O'Connell-Yor polymer and a comparison lemma for the ratios of partition functions. We also show the existence of infinite length limits in the Brownian last passage percolation model.
The problems considered in the present paper have their roots in two different cultures. The 'true' (or myopic) self-avoiding walk model (TSAW) was introduced in the physics literature by Amit, Parisi and Peliti. This is a nearest neighbor non-Markovian random walk in Z^d which prefers to jump to those neighbors which were less visited in the past. The self-repelling Brownian polymer model (SRBP), initiated in the probabilistic literature by Durrett and Rogers (independently of the physics community), is the continuous space-time counterpart: a diffusion in R^d pushed by the negative gradient …
The self-repelling Brownian polymer model (SRBP) initiated by Durrett and Rogers in [Durrett-Rogers (1992)] is the continuous space-time counterpart of the myopic (or 'true') self-avoiding walk model (MSAW) introduced in the physics literature by Amit, Parisi and Peliti in [Amit-Parisi-Peliti (1983)]. In both cases, a random motion in space is pushed towards domains less visited in the past by a kind of negative gradient of the occupation time measure. We investigate the asymptotic behaviour of SRBP in the non-recurrent dimensions. First, extending 1-dimensional results from [Tarres-Toth-Valko…
The Airy line ensemble is a positive-integer indexed system of random continuous curves whose finite dimensional distributions are given by the multi-line Airy process. It is a natural object in the KPZ universality class: for example, its highest curve, the Airy$_2$ process, describes after the subtraction of a parabola the limiting law of the scaled energy of a geodesic running from the origin to a variable point on an anti-diagonal line in such problems as Poissonian last passage percolation. The ensemble of curves resulting from the Airy line ensemble after the subtraction of the same para…
We characterize the law of the partition function of a Brownian directed polymer model in terms of a diffusion process associated with the quantum Toda lattice. The proof is via a multidimensional generalization of a theorem of Matsumoto and Yor concerning exponential functionals of Brownian motion. It is based on a mapping which can be regarded as a geometric variant of the RSK correspondence.
We consider the partition function $Z_{\ell}(\vec x,0\vert \vec y,t)$ of $\ell$ non-intersecting continuous directed polymers of length $t$ in dimension $1+1$, in a white noise environment, starting from positions $\vec x$ and terminating at positions $\vec y$. When $\ell=1$, it is well known that for fixed $x$, the field $\log Z_1(x,0\vert y,t)$ solves the Kardar-Parisi-Zhang equation and admits the Brownian motion as a stationary measure. In particular, as $t$ goes to infinity, $Z_1(x,0\vert y,t)/Z_1(x,0\vert 0,t) $ converges to the exponential of a Brownian motion $B(y)$. In this article, w…
In this paper, we study a model of a Brownian polymer in $\mathbb {R}_+\times \mathbb {R}^d$, introduced by Rovira and Tindel [J. Funct. Anal. 222 (2005) 178--201]. Our investigation focuses mainly on the effect of strong spatial correlation in the environment in that model in terms of free energy, fluctuation exponent and volume exponent. In particular, we prove that under some assumptions, very strong disorder and superdiffusivity hold at all temperatures when $d\ge3$ and provide a novel approach to Petermann's superdiffusivity result in dimension one [Superdiffusivity of directed polymers i…
In this short note, we prove a central limit theorem for a type of replica overlap of the Brownian directed polymer in a Gaussian random environment, in the low temperature regime and in all dimensions. The proof relies on a superconcentration result for the KPZ equation driven by a spatially mollified noise, which is inspired by the recent work of Chatterjee \cite{C1}.
A simple derivation of Spitzer'z asymptotic law for Brownian windings [Trans.Am.Math.Soc.87,187 (1958)]is presented along with its generalizations >.These include the cases of planar Brownian walks interacting with a single puncture and Brownian walks on a single truncated cone with variable conical angle interacting with the truncated conical tip.Such situations are typical in the theories of quantum Hall effect and 2+1 quantum gravity, respectively .They also have some applications in polymer physics
We consider a model of a polymer in $\mathbb{Z}^{d+1}$, constrained to join 0 and a hyperplane at distance $N$. The polymer is subject to a quenched nonnegative random environment. Alternatively, the model describes crossing random walks in a random potential (see Zerner [Ann Appl. Probab. 8 (1998) 246--280] or Chapter 5 of Sznitman [Brownian Motion, Obstacles and Random Media (1998) Springer] for the original Brownian motion formulation). It was recently shown [Ann. Probab. 36 (2008) 1528--1583; Probab. Theory Related Fields 143 (2009) 615--642] that, in such a setting, the quenched and annea…
Motivated by discrete directed polymers in one space and one time dimension, we construct a continuum directed random polymer that is modeled by a continuous path interacting with a space-time white noise. The strength of the interaction is determined by an inverse temperature parameter beta, and for a given beta and realization of the noise the path evolves in a Markovian way. The transition probabilities are determined by solutions to the one-dimensional stochastic heat equation. We show that for all beta > 0 and for almost all realizations of the white noise the path measure has the same Ho…
In the zero temperature Brownian semi-discrete directed polymer we study the joint distribution of two last-passage times at positions ordered in the time-like direction. This is the situation when we have the slow de-correlation phenomenon. We compute the limiting joint distribution function in a scaling limit. This limiting distribution is given by an expansion in determinants which is not a Fredholm expansion. A somewhat similar looking formula was derived non-rigorously in a related model by Dotsenko.
We extend Varadhan's construction of the Edwards polymer model to fractional Brownian loops and fractional Brownian starbursts. We show that, as in the fBm case, the Edwards density under a renormalizaion is an integrable function for the case H<= 1/d.
