Philippe Carmona, Yueyun Hu
In this note we show that in any dimension $d$, the strong disorder property implies the strong localization property. This is established for a continuous time model of directed polymers in a random environment : the parabolic Anderson Model.
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A short proof of diffusivity for the directed polymers in the weak disorder phase
本文针对弱无序相中随机环境下的有向聚合物，给出了扩散性的一个新的简短初等证明。
In this article, we derive strong localization results for directed polymers in random environment. We show that at "low temperature" the polymer measure is asymptotically concentrated at a few points of macroscopic mass (we call these points epsilon-atoms). These results are derived assuming weak conditions on the tail decay of the random environment.
We give an exact expression for the partition function of a continuous time DPRE on a two points state space.
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 ob…
Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should, in most circumstances, be the same as in the random energy model. We show that this conjecture holds true as well for directed polymers in random environment. We also show that, under certain conditions, this conjecture holds for directed polymers even if energy levels that grow moderately with the volume of the system are considered.
We consider two directed polymer models in the Kardar-Parisi-Zhang (KPZ) universality class: the O'Connell-Yor semi-discrete directed polymer with boundary sources and the continuum directed random polymer with (m,n)-spiked boundary perturbations. The free energy of the continuum polymer is the Hopf-Cole solution of the KPZ equation with the corresponding (m,n)-spiked initial condition. This new initial condition is constructed using two semi-discrete polymer models with independent bulk randomness and coupled boundary sources. We prove that the limiting fluctuations of the free energies resca…
We study a model of directed polymers with an exponentially recurrent Markov chain and an indefinitely divisible random environment. We prove that the normalized partition function converges exponentially fast towards zero at all temperatures.
We show that the endpoint large deviation rate function for a continuous-time directed polymer agrees with the rate function of the underlying random walk near the origin in the whole weak disorder phase.
On the localization regime of high-dimensional directed polymers in time-correlated random field
本文研究一般时间相关随机场上的高维定向聚合物，证明了所有温度下大数定律及极限自由能的存在性与平滑性，并通过分离随机场纠缠的技术展示了非局域化-局域化转变。
The intermediate disorder regime for stable directed polymer in Poisson environment
本文研究了1+1维Poisson随机环境中的稳定有向聚合，证明了在中间无序区域的扩散标度下，其归一化点对点配分函数依分布收敛于带分数拉普拉斯算子与高斯乘性噪声的随机热方程的解。
In this note we give upper bounds for the free energy of discrete polymers in random media. The bounds are given by the so-called generalized multiplicative cascades from the statistical theory of turbulence. For the polymer model, we derive that the quenched free energy is different from the annealed one in dimension 1, for any finite temperature and general environment. This implies localization of the polymer.
We consider the convergence of partition functions and endpoint density for the half-space directed polymer model in dimension $1+1$ in the intermediate disorder regime as considered for the full space model by Alberts, Khanin and Quastel in [AKQ]. By scaling the inverse temperature like $βn^{-1/4}$, the point-to-point partition function converges to the chaos series for the solution to stochastic heat equation with Robin boundary condition and delta initial data. We also apply our convergence results to the exact-solvable log-gamma directed polymer model in a half-space.
I discuss models for a continuum directed random polymer in a disordered environment in which the polymer lives on a fractal called the \textit{diamond hierarchical lattice}, a self-similar metric space forming a network of interweaving pathways. This fractal depends on a branching parameter $b\in \mathbb{N}$ and a segmenting number $s\in \mathbb{N}$. For $s>b$ my focus is on random measures on the set of directed paths that can be formulated as a subcritical Gaussian multiplicative chaos. This path measure is analogous to the continuum directed random polymer introduced by Alberts, Khanin, Qu…
For a directed polymer model in random environment, a characterization of the weak disorder phase in terms of the moment of the renormalized partition function has been proved in [S. Junk: Communications in Mathematical Physics 389, 1087-1097 (2022)]. We extend this characterization to a large class of unbounded environments which includes many commonly used distributions.
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.
We study the free energy and its relevant quantity for the directed polymer in random environment. The polymer is allowed to make unbounded jumps and the environment is given by the Bernoulli variables. We first establish the concentration of the ground state energy of polymer at zero temperature. Secondly, we also prove the same property of the free energy at finite temperature. In the proof, we use the fact that the maximum jump of any polymer nearly minimizing energy is not too large with high probability. This is an interesting property itself from the first passage percolation viewpoint.
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.
We study the long-range directed polymer model on $\mathbbm{Z}$ in a random environment, where the underlying random walk lies in the domain of attraction of an $α$-stable process for some $α\in(0,2]$. Similar to the more classic nearest-neighbor directed polymer model, as the inverse temperature $β$ increases, the model undergoes a transition from a weak disorder regime to a strong disorder regime. We extend most of the important results known for the nearest-neighbor directed polymer model on $\mathbbm{Z}^d$ to the long-range model on $\mathbbm{Z}$. More precisely, we show that in the entire…
The sequence of random probability measures $ν_n$ that gives a path of length $n$, $\unsur{n}$ 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.
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 consider the stationary O'Connell-Yor model of semi-discrete directed polymers in a Brownian environment in the intermediate disorder regime and show convergence of the increments of the log-partition function to the energy solutions of the stochastic Burgers equation. The proof does not rely on the Cole-Hopf transform and avoids the use of spectral gap estimates for the discrete model. The key technical argument is a second-order Boltzmann-Gibbs principle.
We consider random walk in a space-time random potential, also known as directed random polymer measures, on the planar square lattice with nearest-neighbor steps and general i.i.d. weights on the vertices. We construct covariant cocycles and use them to prove new results on existence, uniqueness/non-uniqueness, and asymptotic directions of semi-infinite polymer measures (solutions to the Dobrushin-Lanford-Ruelle equations). We also prove non-existence of covariant or deterministically directed bi-infinite polymer measures. Along the way, we prove almost sure existence of Busemann function lim…
We present in this paper the construction of a continuum directed polymer model in an environment given by space-time Lévy noise. One of the main objectives of this construction is to describe the scaling limit of discrete directed polymer in an heavy-tail environment and for this reason we put special emphasis on the case of $α$-stable noises with $α\in (1,2)$. Our construction can be performed in arbitrary dimension, provided that the Lévy measure satisfies specific (and dimension dependent) conditions. We also discuss a few basic properties of the continuum polymer and the relation between …
We prove the scaling relation chi = 2 xi - 1 between the transversal exponent xi and the fluctuation exponent chi for directed polymers in a random environment in d dimensions. The definition of these exponents is similar to that proposed by S. Chatterjee in first-passage percolation. The proof presented here also establishes the relation in the zero temperature version of the model, known as last-passage percolation.
We show that the weak disorder phase for the directed polymer model in a bounded random environment is characterized by the integrability of the running supremum $\sup_{n\in \mathbb N}W_n^β$ of the associated martingale $(W_n^β)_{n\in \mathbb N}$. Using this characterization, we prove that $(W_n^β)_{n\in \mathbb N}$ is $L^p$-bounded in the whole weak disorder phase, for some $p>1$. The argument generalizes to non-negative martingales with a certain product structure.
We study the directed polymer model for general graphs (beyond $\mathbb Z^d$) and random walks. We provide sufficient conditions for the existence or non-existence of a weak disorder phase, of an $L^2$ region, and of very strong disorder, in terms of properties of the graph and of the random walk. We study in some detail (biased) random walk on various trees including the Galton Watson trees, and provide a range of other examples that illustrate counter-examples to intuitive extensions of the $\mathbb Z^d$/SRW result.
In this paper, we consider directed polymers in random environment with discrete space and time. For transverse dimension at least equal to 3, we prove that diffusivity holds for the path in the full weak disorder region, i.e., where the partition function differs from its annealed value only by a non-vanishing factor. Deep inside this region, we also show that the quenched averaged energy has fluctuations of order 1. In complete generality (arbitrary dimension and temperature), we prove monotonicity of the phase diagram in the temperature.
The objective of the present paper is to establish exponential large deviation inequalities, and to use them to show exponential concentration inequalities for the free energy of a polymer in general random environment, its rate of convergence, and an expression of its limit value in terms of those of some multiplicative cascades.
Universality of directed polymers in the intermediate disorder regime
本文证明了Krishnan和Quastel关于一般独立同分布权重有向聚合物在中间无序区域涨落收敛于Tracy--Widom GUE分布的猜想，对任意1/5<α<1/4成立。
In this paper, we consider four integrable models of directed polymers for which the free energy is known to exhibit KPZ fluctuations. A common framework for the analysis of these models was introduced in our recent work on the O'Connell-Yor polymer. We derive estimates for the central moments of the partition function, of any order, on the near-optimal scale $N^{1/3+ε}$, using an iterative method. Among the innovations exploiting the invariant structure, we develop formulas for correlations between functions of the free energy and the boundary weights that replace the Gaussian integration by …
We study the directed polymers in random environment on an infinite graph $G=(V,E)$ on which the underlying random walk satisfies sub-Gaussian heat kernel bounds with spectral dimension $d_{s}$ strictly less than two. Our goal in this paper is to show (i) the existence and the coincidence of the quenched and the annealed free energy $F_q(β)$, $F_a(β)$ and (ii) that $F_a(β)-F_q(β)$ is comparable to $β^{\frac{4}{2-d_{s}}}$ for small inverse temperature $β$.
We consider the model of directed polymers in a random environment introduced by Petermann : the random walk is $\mathbb{R}^d$-valued and has independent gaussian $N(0,I_d)$-increments, and the random media is a stationary centred Gaussian process $(g(k,x), k \geq 1, x \in \mathbb{R}^d)$ with covariance matrix $cov(g(i,x),g(j,y))=δ_{ij} Γ(x-y) ,$ where $Γ$ is a bounded integrable function on $\mathbb{R}^d .$ For this model, we establish an upper bound of the volume exponent in all dimensions $d$.
We study the Cauchy directed polymer model on $\mathbb{Z}^{1+1}$, where the underlying random walk is in the domain of attraction to the $1$-stable law. We show that, if the random walk satisfies certain regularity assumptions and its symmetrized version is recurrent, then the free energy is strictly negative at any inverse temperature $β>0$. Moreover, under additional regularity assumptions on the random walk, we can identify the sharp asymptotics of the free energy in the high temperature limit, namely, \begin{equation*} \lim\limits_{β\to0}β^{2}\log(-p(β))=-c. \end{equation*}
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 $-\exp\left( -\frac{c_\varepsilon}{β^{2+\varepsilon}} \right)$ for small $β$, where $c_\varepsilon>0$ is a constant depending on $\varepsilon>0$. Also, we will suggest a strategy to get a sharper asymptotics.
Non-coincidence of critical points for directed polymers on supercritical percolation clusters
该研究针对d≥3维超临界伯努利边渗流无穷团上的有向聚合物模型，证明其强无序区存在极强无序不成立的非空子相，与全格点上有向聚合物相变的锐度特征形成对比。
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 …
In this article, we consider two models of directed polymers in random environment: a discrete model and a continuous model. We consider these models in dimension greater or equal to 3 and we suppose that the normalized partition function is bounded in L^2. Under these assumptions, Sinai proved a local limit theorem for the discrete model, using a perturbation expansion. In this article, we give a new method for proving Sinai's local limit theorem. This new method can be transposed to the continuous setting in which we prove a similar local limit theorem.
This note sharpens the smoothing inequality of Giacomin and Toninelli for disordered polymers. This inequality is shown to be valid for any disorder distribution with locally finite exponential moments, and to provide an asymptotically sharp constant for weak disorder. A key tool in the proof is an estimate that compares the effect on the free energy of tilting, respectively, shifting the disorder distribution. This estimate holds in large generality (way beyond disordered polymers) and is of independent interest.
We prove that the free energy of directed polymer in Bernoulli environment converges to the growth rate for the number of open paths in super-critical oriented percolation as the temperature tends to zero. Our proof is based on rate of convergence results which hold uniformly in the temperature. We also prove that the convergence rate is locally uniform in the percolation parameter inside the super-critical phase, which implies that the growth rate depends continuously on the percolation parameter.
We consider directed random polymers in $(d+1)$ dimensions with nearly gamma i.i.d. disorder. We study the partition function $Z_{N,ω}$ and establish exponential concentration of $\log Z_{N,ω}$ about its mean on the subgaussian scale $\sqrt{N/\log N}$ . This is used to show that $\mathbb{E}[ \log Z_{N,ω}]$ differs from $N$ times the free energy by an amount which is also subgaussian (i.e. $o(\sqrt{N})$), specifically $O(\sqrt{\frac{N}{\log N}}\log \log N)$.
We study the ergodic theory of stationary directed nearest-neighbor polymer models on $\mathbb Z^2$, with i.i.d. weights. Such models are equivalent to specifying a stationary distribution on the space of weights and correctors that satisfy certain consistency conditions. We show that for prescribed weight distribution and corrector mean, there is at most one stationary polymer distribution which is ergodic under the $e_1$ or $e_2$ shift. Further, if the weights have more than two moments and the corrector mean vector is an extreme point of the superdifferential of the limiting free energy, th…
We consider the discrete directed polymer model with i.i.d. environment and we study the fluctuations of the tail $n^{(d-2)/4}(W_\infty - W_n)$ of the normalized partition function. It was proven by Comets and Liu, that for sufficiently high temperature, the fluctuations converge in distribution towards the product of the limiting partition function and an independent Gaussian random variable. We extend the result to the whole $L^2$-region, which is predicted to be the maximal high-temperature region where the Gaussian fluctuations should occur under the considered scaling. To do so, we manage…
Stability of weak disorder phase for directed polymer with applications to limit theorems
本文证明有向聚合物在弱无序相内部对小偏置扰动具有稳定性，并据此给出中心极限定理的新证明，同时将布朗聚合物模型的收敛改进为几乎必然收敛。
We establish a second-order almost sure limit theorem for the minimal position in a one-dimensional super-critical branching random walk, and also prove a martingale convergence theorem which answers a question of Biggins and Kyprianou [9]. Our method applies furthermore to the study of directed polymers on a disordered tree. In particular, we give a rigorous proof of a phase transition phenomenon for the partition function (from the point of view of convergence in probability), already described by Derrida and Spohn [17]. Surprisingly, this phase transition phenomenon disappears in the sense …
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…
Fractional moments of the Stochastic Heat Flow and 2D Directed Polymers
该研究针对临界二维随机热流与二维定向聚合物模型，通过改进测度变换论证并引入新颖粗粒化程序，给出了在所有参数下均一致的分数阶矩估计，明确其消失由二阶矩发散主导，还提供了定向聚合物的精确二阶矩估计。
In the present work, we investigate the case of Directed Polymer in a Random Environment (DPRE), when the increments of the random walk are heavy-tailed with tail-exponent equal to zero ($\mathbf{P}[|X_1|\geq n]$ decays slower than any power of $n$). This case has not yet been studied in the context of directed polymers and present key differences with the simple symmetric random walk case and the cases where the increments belong to the domain of attraction of an $α$-stable law, where $α\in (0, 2]$. We establish the absence of a very strong disorder regime - that is, the free energy equals ze…
Convergence from the Log-Gamma Polymer to the Directed Landscape
本文通过定义log-gamma片层与景观，利用几何RSK对应和单调性证明其收敛于Airy片层及定向景观，并针对离散模型缺乏尺度不变性发展了新的非对角局部波动估计。
We consider a directed polymer model in dimension $1+1$, where the disorder is given by the occupation field of a Poisson system of independent random walks on $\mathbb Z$. In a suitable continuum and weak disorder limit, we show that the family of quenched partition functions of the directed polymer converges to the Stratonovich solution of a multiplicative stochastic heat equation (SHE) with a Gaussian noise, whose space-time covariance is given by the heat kernel. In contrast to the case with space-time white noise where the solution of the SHE admits a Wiener-Itô chaos expansion, we establ…
The maximum of the two dimensional Gaussian directed polymer in the subcritical regime
该研究针对次临界二维高斯有向聚合物在√N×√N方格上的配分函数最大值展开，证明其与log N的比值收敛于常数σ^*，且该常数与两类已有研究中的对应常数一致。
The free-energy fluctuations of the discrete directed polymer in 1+1 dimensions is conjecturally in the Tracy-Widom universality class at all finite temperatures and in the intermediate disorder regime. Seppäläinen's log-gamma polymer was proven to have GUE Tracy-Widom fluctuations in a restricted temperature range by Borodin et. al. (2013). We remove this restriction, and extend this result into the intermediate disorder regime. This result also identifies the scale of fluctuations of the log-gamma polymer in the intermediate disorder regime, and thus verifies a conjecture of Alberts et. al. …
We study a polymer model on hierarchical lattices very close to the one introduced and studied in \cite{DGr, CD}. For this model, we prove the existence of free energy and derive the necessary and sufficient condition for which very strong disorder holds for all $\gb$, and give some accurate results on the behavior of the free energy at high-temperature. We obtain these results by using a combination of fractional moment method and change of measure over the environment to obtain an upper bound, and second moment method to get a lower bound. We also get lower bounds on the fluctuation exponent…
A note on the asymptotics of the free energy of $1+1$ dimensional directed polymers in random environment at high temperature
本文在不依赖集中不等式的前提下，证明了 $1+1$ 维随机环境中高温定向聚合物自由能的相同尖锐渐近行为，即 $\lim_{β\to0}\frac{1}{β^4}F(β)=-\frac{1}{6}$。
The a.s. existence of a polymer probability in the infinite volume limit is readily obtained under general conditions of weak disorder from standard theory on multiplicative cascades or branching random walk. However, speculations in the case of strong disorder have been mixed. In this note existence of an infinite volume probability is established at critical strong disorder for which one has convergence in probability. Some calculations in support of a specific formula for the a.s. asymptotic variance of the polymer path under strong disorder are also provided.
Long linear polymers in dilute solutions are known to undergo a collapse transition from a random coil (expand itself) to a compact ball (fold itself up) when the temperature is lowered, or the solvent quality deteriorates. A natural model for this phenomenon is a 1+1 dimensional self-interacting and partially directed self-avoiding walk. In this paper, we develop a new method to study the partition function of this model, from which we derive a variational formula for the free energy. This variational formula allows us to prove the existence of the collapse transition and to identify the crit…
Enhanced noise sensitivity, 2D directed polymers and Stochastic Heat Flow
本文将噪声敏感性理论推广至一般随机变量，证明二维有向聚合物配分函数在临界状态具有增强噪声敏感性，并推出随机热流独立于无序白噪声。
The partition function of the directed polymer model on Z^{2+1} undergoes a phase transition in a suitable continuum and weak disorder limit. In this paper, we focus on a window around the critical point. Exploiting local renewal theorems, we compute the limiting third moment of the space-averaged partition function, showing that it is uniformly bounded. This implies that the rescaled partition functions, viewed as a generalized random field on R^2, have non-trivial subsequential limits, and each such limit has the same explicit covariance structure. We obtain analogous results for the stochas…
We consider the model of Directed Polymers in an i.i.d. gaussian or bounded Environment in the $L^2$ region. We prove the convergence of the law of the environment seen by the particle. As a main technical step, we establish a lower tail concentration inequality for the partition function for bounded environments. Our proof is based on arguments developed by Talagrand in the context of the Hopfield Model. This improves in some sense a concentration inequality obtained by Carmona and Hu for gaussian environments. We use this and a Local Limit Theorem to prove the $L^1$ convergence of the densit…
Intermediate disorder for directed polymers with space-time correlations
该研究针对一维乘性噪声随机热方程的聚合物近似问题，提出一类强混合时空环境的广义聚合物模型，采用基于矩的SHE刻画方法证明其收敛于Itô解，收敛拓扑较弱，拓展了环境适用范围。
We discuss variational formulas for the limits of certain models of motion in a random medium: namely, the limiting time constant for last-passage percolation and the limiting free energy for directed polymers. The results are valid for models in arbitrary dimension, steps of the admissible paths can be general, the environment process is ergodic under shifts, and the potential accumulated along a path can depend on the environment and the next step of the path. The variational formulas come in two types: one minimizes over gradient-like cocycles, and another one maximizes over invariant measu…
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