Yu Gu, Tomasz Komorowski
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}.
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The geometric Burge correspondence and the partition function of polymer replicas
本文提出了一种几何Burge对应的构造，展示了其与几何Robinson-Schensted-Knuth对应和Schützenberger反演的关系，并应用该模型证明了聚合物复制配分函数的分布为Whittaker测度。
Recently, Michel Talagrand computed the large deviations limit $\lim_{N\to\infty}(Na)^{-1}\log \e Z_N^a$ for the moments of the partition function $Z_N$ in the Sherrington-Kirkpatrick model for all real $a\geq 0.$ For $a\geq 1$ the limit is given by Guerra's inverse bound and this result extends the classical physicist's replica method that corresponds to integer $a.$ We give a new proof for $a\geq 1$ in the case of the pure $p$-spin SK model that provides a strong exponential control of the overlap.
We study two objects concerning the Wiener sausage among Poissonian obstacles. The first is the asymptotics for the \textit{replica overlap}, which is the intersection of two independent Wiener sausages. We show that it is asymptotically equal to their union. This result confirms that the localizing effect of the media is so strong as to completely determine the motional range of particles. The second is an estimate on the \textit{covering time}. It is known that the Wiener sausage avoiding Poissonian obstacles up to time $t$ is confined in some `clearing' ball near the origin and almost fills…
Certain polymer models are known to exhibit path localization in the sense that at low temperatures, the average fractional overlap of two independent samples from the Gibbs measure is bounded away from $0$. Nevertheless, the question of where along the path this overlap takes place has remained unaddressed. In this article, we prove that on linear scales, overlap occurs along the entire length of the polymer. Namely, we consider time intervals of length $\varepsilon N$, where $\varepsilon>0$ is fixed but arbitrarily small. We then identify a constant number of distinguished trajectories such …
We study the Replica Symmetric region of general multi-species Sherrington-Kirkpatrick (MSK) Model and answer some of the questions raised in Ann.~Probab.~43~(2015), no.~6, 3494--3513, where the author proved the Parisi formula under \emph{positive-definite} assumption on the disorder covariance matrix $Δ^2$. First, we prove exponential overlap concentration at high temperature for both \indf~and \emph{positive-definite} $Δ^2$ MSK model. We also prove a central limit theorem for the free energy using overlap concentration. Furthermore, in the zero external field case, we use a quadratic coupli…
We investigate the high-temperature behavior of the directed polymer model in dimension $1+2$. More precisely we study the difference $Δ\mathtt{F}(β)$ between the quenched and annealed free energies for small values of the inverse temperature $β$. This quantity is associated to localization properties of the polymer trajectories, and is related to the overlap fraction of two replicas. Adapting recent techniques developed by the authors in the context of the disordered pinning model (Berger and Lacoin, arXiv:1503.07315 [math-ph]), we identify the sharp asymptotic high temperature behavior \[\li…
We consider large time behavior of typical paths under the Anderson polymer measure. If $P$ is the measure induced by rate $κ,$ simple, symmetric random walk on $Z^d$ started at $x,$ this measure is defined as $$ dμ(X)={Z^{-1} \exp\{β\int_0^T dW_{X(s)}(s)\}dP(X) $$ where $\{W_x:x\in Z^d\}$ is a field of $iid$ standard, one-dimensional Brownian motions, $β>0, κ>0$ and $Z$ the normalizing constant. We establish that the polymer measure gives a macroscopic mass to a small neighborhood of a typical path as $T \to \infty$, for parameter values outside the perturbative regime of the random walk, giv…
The branching random walk in a uniform magnetic field : magnetization concentration and overlap distributions
本文研究均匀磁场中的高斯二元分支随机游走（BRW），利用一般BRW的已有结果分析其外磁场下的性质，证明低温磁化强度的强集中性，并通过双复制大偏差论证得到Ising超立方体型重叠的分布。
We study the partition function of two versions of the continuum directed polymer in 1+1 dimension. In the full-space version, the polymer starts at the origin and is free to move transversally in the reals, and in the half-space version, the polymer starts at the origin but is reflected at the origin and stays in the negative reals. The partition functions solves the stochastic heat equation in full-space or half-space with mixed boundary condition at the origin; or equivalently the free energy satisfies the Kardar-Parisi-Zhang equation. We derive exact formulas for the Laplace transforms of …
Overlap distribution of spherical spin glass models with general eigenvalue distribution of the interaction matrix
本文研究了具有一般特征值分布的相互作用矩阵的球形自旋玻璃模型，证明了在足够低温下两个样本的重叠分布具有连续密度，揭示了完整的副本破缺特性。
Disorder transfer for the critical SK overlap law
本文针对零场Sherrington-Kirkpatrick模型，在逆温度为一时定量比较不同失序分布，证明在临界重叠极限下将高斯耦合替换为有界对称分布仅引起受控误差，并推广至Wasserstein拓扑下的收敛。
We prove a central limit theorem for the normalized overlap between two replicas in the spherical SK model in the high temperature phase. The convergence holds almost surely with respect to the disorder variables, and the inverse temperature can approach the criticial value at a polynomial rate with any exponent strictly greater than $1/3$.
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