Self-attractive random polymers.
The article studies a repulsion-attraction model for a random polymer of finite length in \( \mathbb{Z}^d \). It is defined as follows; let \( S=(S_i)_{i=0}^n \) be a random variable with \( S_i\in\mathbb{Z}^d \) and \( | S_i-S_{i-1}| =1 \) for \( i=0,\ldots,n, \) where its distribution is derived from a simple random walk, and define a Hamiltonian as a function of \( S \) as \[ H_n(S)=\beta\sum_{i,j}^n1_{\{| S_i-S_j| =0\}}-\frac{\gamma}{2d}\sum_{i,j}^n1_{\{| S_i-S_j| =1\}} \] for some parameters \( \beta,\gamma\geq 0 \). For fixed \( n\in\mathbb{N} \) the Boltzmann distribution \( Q_n^{\beta,\gamma} \) with respect to the Hamiltonian \( H_n(S) \) is called the \(n\) polymer measure with strength of repellence \(\beta \) and strength of attraction \( \gamma \). The authors present three main theorems where the difference between \( \gamma <\beta \) and \( \gamma >\beta \) plays an important role. The first theorem states that for \( \gamma >\beta \) the polymer localizes while it does not if \( \gamma <\beta \). Localization means roughly that there is a finite box such that the probability for the polymer to stay in that box is one in the limit \( n\to\infty \). The second theorem shows that for dimension \( d=1 \) up to translations \( n^{-1}l_n \) converges to some function \( f^{\beta,\gamma} \) in the regime \( \gamma >\beta \), the so-called ``shape'' or ``profile'' of the polymer. Here \( l_n \) denotes the local times for simple random walk, i.e., \(l_n(x)=\sharp\{0\leq i\leq n:S_i=x\}\). Moreover the ``shape'' function \( f^{\beta,\gamma} \) minimizes the Hamiltonian among all functions that occur as limits of rescaled local times \( n^{-1}l_n \). Their third main result is a central limit theorem in the regime \( \gamma < \beta \), where the polymer is not localized, and due to technical reasons the authors have further to take \( \gamma\leq\beta -\frac{1}{2}\log 2\). Then they show \[ \lim_{n\to\infty} Q_n^{\beta,\gamma}\left(\frac{| S_n| -\theta n}{\sigma\sqrt{n}}\in \cdot\right)=\mathcal{N}(0,1), \] for \( \theta\in (0,1] \) and \( \sigma\in (0,\infty)\).
Knight's theorem for local times of simple random walk
scientific article; zbMATH DE number 1906222
scientific article