Subsequential limit

Abstract: The uniform spanning tree (UST) and the loop-erased random walk (LERW) are 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 the 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 2 dimensions. 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 Loewner differential equation ${\partial f\over\partial t} = z {\zeta(t)+z \over \zeta(t)-z} {\partial f\over\partial z}$ with boundary values $f(z,0)=z$, in the range $z\in\U=\{w\in\C\st |w|<1\}$, $t\le 0$. We choose $\zeta(t):= \B(-2t)$, where $\B(t)$ is Brownian motion on $\partial \U$ starting at a random-uniform point in $\partial \U$. Assuming the conformal invariance of the LERW scaling limit in the plane, we prove that the scaling limit of LERW from 0 to $\partial\U$ has the same law as that of the path $f(\zeta(t),t)$. We believe that a variation of this process gives the scaling limit of the boundary of macroscopic critical percolation clusters.

Abstract: Let K be a compact subset of R, or, more generally, of an (n+1)-dimensional riemannian manifold. We suppose that K is mean-convex. If the boundary of K is smooth and connected, this means that the mean curvature of ∂K is everywhere nonnegative (with respect to the inward unit normal) and is not identically 0. More generally, it means that Ft(K) is contained in the interior of K for t > 0, where Ft(K) is the set obtained by letting K evolve for time t under the level set mean curvature flow. As K evolves, it traces out a closed set K of spacetime: K = {(x, t) ∈ R ×R : x ∈ Ft(K)}. Also, there is associated to K a Brakke flow M : t 7→Mt of rectifiable varifolds. We call the pair (M,K) a mean-convex flow. Let X = (x, t) be a point in spacetime with t > 0. Suppose (xi, ti) is a sequence of points converging to X and λi is a sequence of positive numbers tending to infinity. Translate the pair M and K in spacetime by (y, τ) 7→ (y − xi, τ − ti) and then dilate parabolically by (y, τ) 7→ (λiy, λi τ) to get new flows Mi and Ki. The sequence (Mi,Ki) is called a blow-up sequence at X . General compactness theorems guarantee that this sequence will have subsequential limits. A subsequential limit (M′,K′) is called a limit flow. Here M′ : t ∈ (−∞,∞) 7→M ′ t

Abstract: The existence of the weak limit as n → ∞ of the uniform measure on rooted triangulations of the sphere with n vertices is proved. Some properties of the limit are studied. In particular, the limit is a probability measure on random triangulations of the plane.

Abstract: Convergence analysis is carried out for a forward-backward splitting/generalized gradient projection method for the minimization of a special class of non-smooth and genuinely non-convex minimization problems in infinite-dimensional Hilbert spaces. The functionals under consideration are the sum of a smooth, possibly non-convex and non-smooth, necessarily non-convex functional. For separable constraints in the sequence space, we show that the generalized gradient projection method amounts to a discontinuous iterative thresholding procedure, which can easily be implemented. In this case we prove strong subsequential convergence and moreover show that the limit satisfies strengthened necessary conditions for a global minimizer, i.e., it avoids a certain set of non-global minimizers. Eventually, the method is applied to problems arising in the recovery of sparse data, where strong convergence of the whole sequence is shown, and numerical tests are presented.