标题： 强关联随机环境中的极小曲面 显示英文标题

标题： Minimal surfaces in strongly correlated random environments

摘要： 随机环境中极小曲面（MSRE）是在$(d+n)$维空间中的$d$维曲面，它在规定的边界值条件下最小化其弹性能量与环境势能之和。除了自身的内在兴趣外，这类曲面还因其与无序自旋系统以及第一通量渗流模型的联系而受到进一步关注。 在这项工作中，我们研究了强相关环境下的情况，这种环境由分数布朗环境中调和 MSRE 模型实现，其中 Hurst 参数为$H\in(0,1)$。这包括布朗环境的情况（$H=1/2$和$n=1$），后者常用于近似$(d+1)$维随机场伊辛模型的域壁。 我们证明了$d\in\{1,2,3\}$维表面会以幂律波动而离域，并确定了它们精确的横向和最小能量波动指数，以及控制其分布尾部衰减的拉伸指数。 这些指数在所有余维数 $n$ 中均发现是相同的，仅依赖于 $d$ 和 $H$。 横截面和最小能量涨落指数由两个标度关系确定。 我们进一步表明，维度为 $d=4$ 的表面以亚幂律涨落而脱离局域化，其高度和最小能量涨落由一个标度关系联系在一起。 最后，我们证明了维度为 $d\ge 5$ 的表面会局域化。 这些结果将物理学文献中的若干预测建立在严格的数学基础上。 显示英文摘要

摘要： A minimal surface in a random environment (MSRE) is a $d$-dimensional surface in $(d+n)$-dimensional space which minimizes the sum of its elastic energy and its environment potential energy, subject to prescribed boundary values. Apart from their intrinsic interest, such surfaces are further motivated by connections with disordered spin systems and first-passage percolation models. In this work, we consider the case of strongly correlated environments, realized by the model of harmonic MSRE in a fractional Brownian environment of Hurst parameter $H\in(0,1)$. This includes the case of Brownian environment ($H=1/2$ and $n=1$), which is commonly used to approximate the domain walls of the $(d+1)$-dimensional random-field Ising model. We prove that surfaces of dimension $d\in\{1,2,3\}$ delocalize with power-law fluctuations, and determine their precise transversal and minimal energy fluctuation exponents, as well as the stretched exponential exponents governing the tail decay of their distributions. These exponents are found to be the same in all codimensions $n$, depending only on $d$ and $H$. The transversal and minimal energy fluctuation exponents are specified by two scaling relations. We further show that surfaces of dimension $d=4$ delocalize with sub-power-law fluctuations, with their height and minimal energy fluctuations tied by a scaling relation. Lastly, we prove that surfaces of dimensions $d\ge 5$ localize. These results put several predictions from the physics literature on mathematically rigorous ground.