Polymers In Random Environment Research Articles

The critical 2d Stochastic Heat Flow

We consider directed polymers in random environment in the critical dimension d = 2, focusing on the intermediate disorder regime when the model undergoes a phase transition. We prove that, at criticality, the diffusively rescaled random field of partition functions has a unique scaling limit: a universal process of random measures on {mathbb {R}}^2 with logarithmic correlations, which we call the Critical 2d Stochastic Heat Flow. It is the natural candidate for the long sought solution of the critical 2d Stochastic Heat Equation with multiplicative space-time white noise.

Another look at the Balázs-Quastel-Seppäläinen theorem

We study the KPZ equation with a 1 + 1 − 1+1- dimensional spacetime white noise, started at equilibrium, and give a different proof of the main result of Balázs, Quastel, and Seppäläinen [J. Amer. Math. Soc. 24 (2011), pp. 683–708], i.e., the variance of the solution at time t t is of order t 2 / 3 t^{2/3} . Instead of using a discrete approximation through the exclusion process and the second class particle, we utilize the connection to directed polymers in random environment. Along the way, we show the annealed density of the stationary continuum directed polymer equals to the two-point covariance function of the stationary stochastic Burgers equation, confirming the physics prediction of Maes and Thiery [J. Stat. Phys. 168 (2017), pp. 937–963].

Weak Coupling Limits for Directed Polymers in Tube Environments

In this paper, we study a model of directed polymers in random environment, where the environment is restricted to a time-space tube whose spatial width grows polynomially with time. It can be viewed as an interpolation between the disordered pinning model and the classic directed polymer model. We prove weak coupling limits for the directed polymer partition functions in such tube environments in all dimensions, as the inverse temperature vanishes at a suitable rate. As the tube width varies, transitions between regimes of disorder irrelevance, marginal relevance and disorder relevance are observed.

Undirected Polymers in Random Environment: path properties in the mean field limit.

We consider the problem of undirected polymers (tied at the endpoints) in random environment, also known as the unoriented first passage percolation on the hypercube, in the limit of large dimensions. By means of the multiscale refinement of the second moment method we obtain a fairly precise geometrical description of optimal paths, i.e. of polymers with minimal energy. The picture which emerges can be loosely summarized as follows. The energy of the polymer is, to first approximation, uniformly spread along the strand. The polymer's bonds carry however a lower energy than in the directed setting, and are reached through the following geometrical evolution. Close to the origin, the polymer proceeds in oriented fashion -- it is thus as stretched as possible. The tension of the strand decreases however gradually, with the polymer allowing for more and more backsteps as it enters the core of the hypercube. Backsteps, although increasing the length of the strand, allow the polymer to connect reservoirs of energetically favorable edges which are otherwise unattainable in a fully directed regime. These reservoirs lie at mesoscopic distance apart, but in virtue of the high dimensional nature of the ambient space, the polymer manages to connect them through approximate geodesics with respect to the Hamming metric: this is the key strategy which leads to an optimal energy/entropy balance. Around halfway, the mirror picture sets in: the polymer tension gradually builds up again, until full orientedness close to the endpoint. The approach yields, as a corollary, a constructive proof of the result by Martinsson [Ann. Appl. Prob. 26 (2016), Ann. Prob. 46 (2018)] concerning the leading order of the ground state.

A PDE hierarchy for directed polymers in random environments

For a Brownian directed polymer in a Gaussian random environment, with q(t, ⋅) denoting the quenched endpoint density and we derive a hierarchical PDE system satisfied by . We present two applications of the system: (i) we compute the generator of for some special functionals, where is viewed as a Markov process taking values in the space of probability measures; (ii) in the high temperature regime with d ⩾ 3, we prove a quantitative central limit theorem for the annealed endpoint distribution of the diffusively rescaled polymer path. We also study a nonlocal diffusion-reaction equation motivated by the generator and establish a super-diffusive O(t 2/3) scaling.

The endpoint distribution of directed polymers

Probabilistic models of directed polymers in random environment have received considerable attention in recent years. Much of this attention has focused on integrable models. In this paper, we introduce some new computational tools that do not require integrability. We begin by defining a new kind of abstract limit object, called “partitioned subprobability measure,” to describe the limits of endpoint distributions of directed polymers. Inspired by a recent work of Mukherjee and Varadhan on large deviations of the occupation measure of Brownian motion, we define a suitable topology on the space of partitioned subprobability measures and prove that this topology is compact. Then using a variant of the cavity method from the theory of spin glasses, we show that any limit law of a sequence of endpoint distributions must satisfy a fixed point equation on this abstract space, and that the limiting free energy of the model can be expressed as the solution of a variational problem over the set of fixed points. As a first application of the theory, we prove that in an environment with finite exponential moment, the endpoint distribution is asymptotically purely atomic if and only if the system is in the low temperature phase. The analogous result for a heavy-tailed environment was proved by Vargas in 2007. As a second application, we prove a subsequential version of the longstanding conjecture that in the low temperature phase, the endpoint distribution is asymptotically localized in a region of stochastically bounded diameter. All our results hold in arbitrary dimensions, and make no use of integrability.

Zero temperature limit for the Brownian directed polymer among Poissonian disasters

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.

From directed polymers in spatial-correlated environment to stochastic heat equations driven by fractional noise in [formula omitted] dimensions

We consider the limit behavior of partition function of directed polymers in random environment, which is represented by a linear model instead of a family of i.i.d.variables in 1+1 dimensions. Under the assumption on the environment that its spatial correlation decays algebraically, using the method developed in Alberts et al. (2014), we show that the scaled partition function, as a process defined on [0,1]×R, converges weakly to the solution to some stochastic heat equations driven by fractional Brownian field. The fractional Hurst parameter is determined by the correlation exponent of the random environment. Here multiple Itô integral with respect to fractional Gaussian field and spectral representation of stationary process are heavily involved.

Proof of the Path Localization Conjecture for Directed Polymers

It is a well-known open problem in the literature on random polymers to show that a directed polymer in random environment localizes around a favorite path at low temperature. A precise statement of this conjecture is formulated and proved in this article.

Free energy of directed polymers in random environment in $1+1$-dimension at high temperature

We consider the free energy $F(\beta )$ of the directed polymers in random environment in $1+1$-dimension. It is known that $F(\beta )$ is of order $-\beta ^4$ as $\beta \to 0$ [3, 28, 42]. In this paper, we will prove that under a certain dimension free concentration condition on the potential, \[ \lim _{\beta \to 0}\frac{F(\beta )} {\beta ^4}=\lim _{T\to \infty }\frac{1} {T}P_\mathcal{Z} \left [\log \mathcal{Z} _{\sqrt{2} }(T)\right ] =-\frac{1} {6}, \] where $\{\mathcal{Z} _\beta (t,x):t\geq 0,x\in \mathbb{R} \}$ is the unique mild solution to the stochastic heat equation \[ \frac{\partial } {\partial t}\mathcal{Z} =\frac{1} {2}\Delta \mathcal{Z} +\beta \mathcal{Z} {\dot{\mathcal W} }, \lim _{t\to 0}\mathcal{Z} (t,x)dx=\delta _{0}(dx), \] where $\mathcal{W} $ is a time-space white noise and \[ \mathcal{Z} _\beta (t)=\int _\mathbb{R} \mathcal{Z} _\beta (t,x)dx. \]

Localization of directed polymers with general reference walk

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 $\alpha$-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.

Rate of convergence for polymers in a weak disorder

We consider directed polymers in random environment on the lattice Zd at small inverse temperature and dimension d≥3. Then, the normalized partition function Wn is a regular martingale with limit W. We prove that n(d−2)/4(Wn−W)/Wn converges in distribution to a Gaussian law. Both the polynomial rate of convergence and the scaling with the martingale Wn are different from those for polymers on trees.

Limiting Results for the Free Energy of Directed Polymers in Random Environment with Unbounded Jumps

We study asymptotics of the free energy for the directed polymer in random environment. The polymer is allowed to make unbounded jumps and the environment is given by Bernoulli variables. We first establish the existence and continuity of the free energy including the negative infinity value of the coupling constant $\beta$. Our proof of existence at $\beta=-\infty$ differs from existing ones in that it avoids the direct use of subadditivity. Secondly, we identify the asymptotics of the free energy at $\beta=-\infty$ in the limit of the success probability of the Bernoulli variables tending to one. It is described by using the so-called time constant of a certain directed first passage percolation. Our proof relies on a certain continuity property of the time constant, which is of independent interest.

A remark on the bound for the free energy of directed polymers in random environment in 1+2 dimension

We consider the behavior of the quantity p(β); the free energy of directed polymers in random environment in 1 + 2 dimension, where β is inverse temperature. We know that the free energy is strictly negative when β is not zero. In this paper, we will prove that p(β) is bounded from above by \documentclass[12pt]{minimal}\begin{document}$-\exp \left(-\frac{c_\varepsilon }{\beta ^{2+\varepsilon }}\right)$\end{document}−exp−cɛβ2+ɛ for small β, where cε > 0 is a constant depending on ε > 0. Also, we will suggest a strategy to get a sharper asymptotics. We denote by \documentclass[12pt]{minimal}\begin{document}$(\Omega , {\cal F},P )$\end{document}(Ω,F,P) a probability space.

Fluctuation exponents for directed polymers in the intermediate disorder regime

We derive exact fluctuation exponents for a solvable model of one-dimensional directed polymers in random environment in the intermediate scaling regime. This regime corresponds to taking the inverse temperature to zero as the size of the system goes to infinity. The exponents satisfy the KPZ scaling relation and coincide with physical predictions.In the critical case, we recover the fluctuation exponent of the Hopf-Cole solution of the KPZ equation in equilibrium and close to equilibrium.

Localization Transition for Polymers in Poissonian Medium

We study a model of directed polymers in random environment in dimension $1+d$, given by a Brownian motion in a Poissonian potential. We study the effect of the density and the strength of inhomogeneities, respectively the intensity parameter $\nu$ of the Poisson field and the temperature inverse $\beta$. Our results are: (i) fine information on the phase diagram, with quantitative estimates on the critical curve; (ii) pathwise localization at low temperature and/or large density; (iii) complete localization in a favourite corridor for large $\nu \beta^2$ and bounded $\beta$.

The critical branching random walk in a random environment dies out

We study the possibility for branching random walks in random environment (BRWRE) to survive. The particles perform simple symmetric random walks on the $d$-dimensional integer lattice, while at each time unit, they split into independent copies according to time-space i.i.d. offspring distributions. As noted by Comets and Yoshida, the BRWRE is naturally associated with the directed polymers in random environment (DPRE), for which the quantity $\Psi$ called the free energy is well studied. Comets and Yoshida proved that there is no survival when $\Psi 0$. We proved here that, except for degenerate cases, the BRWRE always die when $\Psi=0$. This solves a conjecture of Comets and Yoshida.

Sharp asymptotics for the free energy of 1+1 dimensional directed polymers in an infinitely divisible environment

We give sharp estimate for the free energy of directed polymers in random environment in dimension 1+1. This estimate was known for a Gaussian environment, we extend it to the case where the law of the environment is infinitely divisible.

Influence of spatial correlation for directed polymers

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 in random environment (2000) Ph.D. thesis]. We also derive results for a Brownian model of pinning in a nonrandom potential with power-law decay at infinity.

Directed polymer in random environment and last passage percolation

The sequence of random probability measures νn that gives a path of length n, times the sum of the random weights collected along the paths, is shown to satisfy a large deviations principle with good rate function the Legendre transform of the free energy of the associated directed polymer in a random environment. Consequences on the asymptotics of the typical number of paths whose collected weight is above a fixed proportion are then drawn.