Existence Of Scaling Limit Research Articles

Scalability of Phase Change Materials in Nanostructure Template

The scalability of In2Se3, one of the phase change materials, is investigated. By depositing the material onto a nanopatterned substrate, individual In2Se3nanoclusters are confined in the nanosize pits with well-defined shape and dimension permitting the systematic study of the ultimate scaling limit of its use as a phase change memory element. In2Se3of progressively smaller volume is heated inside a transmission electron microscope operating in diffraction mode. The volume at which the amorphous-crystalline transition can no longer be observed is taken as the ultimate scaling limit, which is approximately 5 nm3for In2Se3. The physics for the existence of scaling limit is discussed. Using phase change memory elements in memory hierarchy is believed to reduce its energy consumption because they consume zero leakage power in memory cells. Therefore, the phase change memory applications are of great importance in terms of energy saving.

Phase structure of non-commutative scalar field theories

We investigate the phase structure of non-commutative scalar field theories and find evidence for ordered phases which break translation invariance. A self-consistent one-loop analysis indicates that the transition into these ordered phases is first order. The phase structure and the existence of scaling limits provides an alternative to the structure of counter-terms in determining the renormalizability of non-commutative field theories. On the basis of the existence of a critical point in the closely related planar theory, we argue that there are renormalizable interacting non-commutative scalar field theories in dimensions two and above. We exhibit this renormalization explicitly in the large $N$ limit of a non-commutative O(N) vector model.

Scaling limits of loop-erased random walks and uniform spanning trees

The uniform spanning tree (UST) and the loop-erased random walk (LERW) are strongly related probabilistic processes. We consider the limits of these models on a fine grid in the plane, as the mesh goes to zero. Although the existence of scaling limits is still unproven, subsequential scaling limits can be defined in various ways, and do exist. We establish some basic a.s. properties of these subsequential scaling limits in the plane. It is proved that any LERW subsequential scaling limit is a simple path, and that the trunk of any UST subsequential scaling limit is a topological tree, which is dense in the plane.The scaling limits of these processes are conjectured to be conformally invariant in dimension 2. We make a precise statement of the conformal invariance conjecture for the LERW, and show that this conjecture implies an explicit construction of the scaling limit, as follows. Consider the Löwner differential equation $$\frac{{\partial f}}{{\partial t}} = z\frac{{\zeta (t) + z}}{{\zeta (t) - z}}\frac{{\partial f}}{{\partial z}}$$ (1) , with boundary valuesf(z,0)=z, in the rangez ∈U= {w ∈ ℂ : •w• < 1},t≤0. We choose ζ(t):=B(−2t), where B(t) is Brownian motion on ∂\( \mathbb{U} \) starting at a random-uniform point in ∂\( \mathbb{U} \). Assuming the conformal invariance of the LERW scaling limit in the plane, we prove that the scaling limit of LERW from 0 to ∂\( \mathbb{U} \) has the same law as that of the pathf(ζ(t),t) (wheref(z,t) is extended continuously to ∂\( \mathbb{U} \)) ×(−∞,0]). We believe that a variation of this process gives the scaling limit of the boundary of macroscopic critical percolation clusters.