Directed polymer in random environment and two points state space

Strong disorder for a certain class of directed polymers in a random environment

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.

On large deviation rate functions for a continuous-time directed polymer in weak disorder

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.

关于时间相关随机场中高维定向聚合物的局域化区域

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。

Poisson环境中稳定有向聚合的中间无序区域

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。

Majorizing multiplicative cascades for directed polymers in random media

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.

Intermediate Disorder regime for half-space directed polymers

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.

Continuum directed random polymers on disordered hierarchical diamond lattices

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…

Moment characterization of the weak disorder phase for directed polymers in a class of unbounded environments

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.

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.

Concentration results for directed polymer with unbounded jumps

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.

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.

On the Long-range Directed Polymer Model

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…

Directed polymer in random environment and last passage percolation

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.

On the Brownian directed polymer in a Gaussian random environment

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.

Stationary Directed Polymers and Energy Solutions of the Burgers Equation

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.

Busemann functions and Gibbs measures in directed polymer models on $\mathbb{Z}^2$

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…

The continuum directed polymer in Lévy Noise

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 …

The scaling relation chi = 2 xi - 1 for directed polymers in a random environment

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.

New characterization of the weak disorder phase of directed polymers in bounded random environments

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.

Directed polymers on infinite graphs

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.

Directed Polymers in Random Environment are Diffusive at Weak Disorder

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.

Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in random environment

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.

中间无序区域中有向聚合物的普适性

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。

Concentration for integrable directed polymer models

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 …

Two-sided bounds on free energy of directed polymers on strongly recurrent graphs

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 $β$.

Upper bound of a volume exponent for directed polymers in a random environment

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 $d$.

Free Energy of the Cauchy Directed Polymer Model at High Temperature

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*}

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 $-\exp\left( -\frac{c_\varepsilon}{β^{2+\varepsilon}} \right)$ 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.

超临界渗流团上有向聚合物临界点的非重合性

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。

Asymmetric directed polymers in random environments

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 …

A Local limit theorem for directed polymers in random media: the continuous and the discrete case

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.

Number of paths in oriented percolation as zero temperature limit of directed polymer

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.

Subgaussian concentration and rates of convergence in directed polymers

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 $O(\sqrt{\frac{N}{\log N}}\log \log N)$.

Uniqueness and ergodicity of stationary directed polymers on $\mathbb Z^2$

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…

Gaussian fluctuations for the directed polymer partition function for $d\geq 3$ and in the whole $L^2$-region

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…

有向聚合物弱无序相的稳定性及其在极限定理中的应用

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。

Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees

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 …

Sharp asymptotics for the partition function of some continuous-time directed polymers

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…

临界二维随机热流与二维定向聚合物的分数阶矩

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。

Directed Polymer for very heavy tailed random walks

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…

从Log-Gamma聚合物到定向景观的收敛性

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。

Scaling limit of a directed polymer among a Poisson field of independent walks

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…

次临界区域下二维高斯有向聚合物的最大值

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。

Directed Polymers on Hierarchical Lattices with site disorder

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…

关于 $1+1$ 维随机环境中高温下定向聚合物自由能渐近性的注记

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 $\lim_{β\to0}\frac{1}{β^4}F(β)=-\frac{1}{6}$。

A variational formula for the free energy of the partially directed polymer collapse

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…

增强噪声敏感性、二维有向聚合物与随机热流

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。

On the moments of the (2+1)-dimensional directed polymer and stochastic heat equation in the critical window

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…

Convergence of the law of the Environment Seen by the Particle for Directed Polymers in Random Media in the $L^2$ region

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…

具有时空相关性的有向聚合物的中间无序性

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。

Variational formulas and cocycle solutions for directed polymer and percolation models

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…

中间无序区有向聚合物的改进普适性界

Improved universality bounds for directed polymers in the intermediate disorder regime

我们通过改进Lindeberg替换方法，将中间无序区有向聚合物的Tracy-Widom GUE普适性范围扩展到更广的逆温度参数区间。

关于二维有向聚合物对数配分函数的中心极限定理

On the Central Limit Theorem for the log-partition function of 2D directed polymers

本文针对二维有向聚合物在次临界温度下对数配分函数的中心极限定理，利用解耦论证和独立随机变量之和的 CLT，并借助与分支随机游走的类比，给出了一个初等证明。

通过矩方法研究临界随机热流（SHF）的连续时间定向聚合物

Continuous-time directed polymers to the Critical SHF via moments

本文针对关联布朗环境中的二维连续时间定向随机聚合物，利用矩方法证明其配分函数收敛到临界随机热流（SHF），为临界SHF的离散近似提供了收敛性理论支撑。