论文标题
无序的广义波兰 - 雪花agaga模型的缩放限制DNA变性模型
Scaling limit of the disordered generalized Poland--Scheraga model for DNA denaturation
论文作者
论文摘要
1970年代引入的波兰 - 切尔加拉模型是描述DNA变性转变的参考模型。最近，它已被概括，以允许在链长度和循环形成中不对称：数学表示基于双变量续订过程，描述了将粘结在一起的碱基对。在本文中，我们考虑了该模型的无序版本，其中两条线通过潜在的$βV（\hatΩ_i，\barΩ_j）+h $相互作用。在这里，$ h \ in \ mathbb r $是一个均匀的固定参数，$（\hatΩ_i）_ {i \ geq 1} $和$（\barΩ_j）_ {j \ geq 1} $是i.i.d. i.i.d. i.i.d.〜随机变量的两个序列，每个dna is in is $ v（$ v（$ v） $β> 0 $是疾病强度。我们的主要结果发现了基础双变量续订的某种条件，因此，如果一个人采用$β，则以某种适当的（明确）速率h \ downarrow0 $，因为链的长度到达无穷大，该模型的分区函数承认，非琐碎的，即无序的缩放限制。这被称为\ textit {中间疾病}制度，与与变性过渡相关的障碍问题有关。有趣的是，令人惊讶的是，必须采用$β\ downarrow0 $的速率取决于交互函数$ v（\ cdot，\ cdot）$以及$（\hatΩ_i）_ {i \ geq 1} $，$（\barΩ_j）_ {\barΩ_j）_ {j \ geq 1} $的分布。另一方面，分区函数的中间混乱限制（存在）是普遍的：它表示为迭代积分的混乱扩展到高斯流程〜$ \ nathcal {m Mathcal {m} $，它作为字段的缩放限制$（e^^βv（e^βV（e^hat（hat hat hat（\ hat hat））线条和列上的强相关性。
The Poland--Scheraga model, introduced in the 1970's, is a reference model to describe the denaturation transition of DNA. More recently, it has been generalized in order to allow for asymmetry in the strands lengths and in the formation of loops: the mathematical representation is based on a bivariate renewal process, that describes the pairs of bases that bond together. In this paper, we consider a disordered version of the model, in which the two strands interact via a potential $βV(\hatω_i,\barω_j)+h$ when the $i$-th monomer of the first strand and the $j$-th monomer of the second strand meet. Here, $h\in\mathbb R$ is a homogeneous pinning parameter, $(\hatω_i)_{i\geq 1}$ and $(\barω_j)_{j\geq 1}$ are two sequences of i.i.d.~random variables attached to each DNA strand, $V(\cdot,\cdot)$ is an interaction function and $β>0$ is the disorder intensity. Our main result finds some condition on the underlying bivariate renewal so that, if one takes $β,h\downarrow0$ at some appropriate (explicit) rate as the length of the strands go to infinity, the partition function of the model admits a non-trivial, i.e. disordered, scaling limit. This is known as an \textit{intermediate disorder} regime and is linked to the question of disorder relevance for the denaturation transition. Interestingly and surprisingly, the rate at which one has to take $β\downarrow0$ depends on the interaction function $V(\cdot,\cdot)$ and on the distribution of $(\hatω_i)_{i\geq 1}$, $(\barω_j)_{j\geq 1}$. On the other hand, the intermediate disorder limit of the partition function, when it exists, is universal: it is expressed as a chaos expansion of iterated integrals against a Gaussian process~$\mathcal{M}$, which arises as the scaling limit of the field $(e^{βV(\hatω_i,\barω_j)})_{i,j\geq 0}$ and exhibits strong correlations on lines and columns.
