Limit theorems are established in three different contexts. The first one concerns exceptional points of the simple random walk in planar lattice domains approximating a given bounded continuum domain D of R^2 with wired boundary conditions; the walk is run for a time proportional to the expected cover time. The sets of suitably defined thick, thin, light and avoided points are shown to be asymptotically distributed according to a log-normal multiple of the zero-average Liouville Quantum Gravity measure in D.

The second area of interest concerns the scaling limit of 2-neighbor polluted bootstrap percolation on Z^2. Here each site is initially independently declared polluted with probability q, occupied with probability p, and vacant otherwise. At each step, each vacant site becomes occupied by contact with 2 or more occupied neighbors. It is shown that in the limit when p, q ↓ 0 with q/p^2 → λ [0,∞) , the regime of small λ results in asymptotic full occupancy while the regime of large \lambda results in asymptotic full vacancy of the terminal configuration. The proof is based on an identification of a continuum percolation model of blocking contours where these regimes correspond to absence and presence of ordinary percolation, respectively.

The last area of interest concerns the number of descents and peaks in a given conjugacy class of a random permutation of n elements. Asymptotic normality is proved proved in the limit n→∞ for suitably scaled versions of these quantities by establishing a uniform estimate on their moment generating functions.

This thesis is concerned with scaling limits of sequences of random isoperimetric problems. We first consider progressively larger isoperimetric subgraphs of the infinite cluster $\textbf{C}_\infty$ of supercritical bond percolation on $\Z^d$ for $d \geq 3$. We prove a shape theorem for these subgraphs, showing that upon rescaling they tend almost surely to a deterministic shape, which is itself an isoperimetric set for a norm we construct. The norm represents a homogenized surface energy arising from random interfaces between subgraphs of $\textbf{C}_\infty$. We obtain sharp asymptotics for a modification of the Cheeger constant of $\textbf{C}_\infty \cap [-n,n]^d$, settling a conjecture of Benjamini for the version of the Cheeger constant defined here.

We also study the isoperimetric properties of the giant component in dimension two using the original definition of the Cheeger constant, taking into account the boundary of the large box $[-n,n]^d$. Analogous results are shown here, with the caveat that a more complicated continuum isoperimetric problem emerges due to the presence of the boundary.

Limit theorems are ubiquitous in probability theory. The present work samples contributions

of the author at the interface of this theory with three distinct fields: interacting particle

systems, exchangeable random variables, and long-range percolation.

In the theory of interacting particle systems, one often studies the stationary distribu

tions, which are obtained as limiting distributions of the process. We will discuss a proof

concerning characterization of these measures in the case of an attractive nearest neighbor

translation invariant spin system on the integers.

Exchangeable sequences of random variables are mixtures of i.i.d. sequences, and the

probability measure that determines the relative proportions of this mixture can be ob

tained as a limit from the exchangeable sequence itself. We will analyze the possibility of

reconstructing this probability measure from only partial information about the exchange

able sequence.

A goal in long-range percolation is to understand how chemical distance scales with

Euclidean separation. We will show that the limiting scaling behavior for a certain class of

models is polylogarithmic. This will be an improvement on existing results.

We consider the continuity equation for a population density subject to (i) a density upper-bound that depends on space and time and (ii) a velocity that minimizes the kinetic energy. A solution is constructed via the Wasserstein minimizing movement scheme for a corresponding time-dependent energy. Motion of the solution is driven by a decreasing density constraint.

With a few assumptions, we prove this solution moves according to a free boundary problem of modified Hele-Shaw type that depends on the density constraint. In order to do this, we utilize a modified porous medium equation as an approximation to the original problem. Viscosity solution arguments are used to prove that given a decreasing density constraint, the porous medium equation solutions converge to the Hele-Shaw free boundary problem solution. By analyzing the Wasserstein gradient flow structure of the time-dependent energies involved, we next show that the porous medium equation solutions also converge to a solution of the original problem, thus identifying it with the Hele-Shaw description.

In addition, we consider complications of the analysis without each assumption, perform numerical simulations supporting the results, and explore some limiting situations of the dynamics.