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The scaling exponent for the diameter should be $\frac{5}{4}$. Indeed, it follows from Wilson's algorithm that a branch of a uniform random spanning tree, say between two given vertices, is a loop-era …

Consider a similar question for Brownian motion: $$F(T)=\mathbb{P}(|B_t|\leq \sqrt{t}\quad \forall 1\leq t\leq T).$$ Then, we have $$F(2^N)\leq \mathbb{P}(|B_{t}-B_{2^n}|\leq 2\sqrt{2^{n+1}} \quad \fo …

There are several very different cases here. The first case is when $\mathbb{E}(X_i)=\mathbb{E}(Y_i)=0$, which in your case means that the variables are symmetric. In this case, $P(\sum_{i=0}^{kn} X_i …

To give a specific reference, T1 in Section 8, Chapter II of Spitzer: Principles of random walks asserts that a 2D random walk is recurrent if $\sum_{x\in\mathbb{Z}^2}|x^2|P(0,x)<\infty$, where $P$ ar …

As the question is asked, the answer is "no": if a continuous curve $\gamma:\mathbb{R}_{\geq 0}\to [0,1)^2$ is self-avoiding, i.e., injective, then the image $\gamma(\mathbb{R}_{\geq 0})$ is nowhere d …

What fails for other lattices is that there seems to be no parafermionic observable with properties as nice as for hexagonal lattice; specifically, there is no analog of Lemma 1. In the definition of …

See Jacod, Shiryaev, "LImit theorems for stochastic processes", Chapter IX.

The Cramer-Esseen theorem (Theorem 2 in paragraph 42 of Gnedenko--Kolmogorov book) seems to give a very sharp asymptotics for the terms in your sum in the linked answer, at least for large $n$. Now, f …

Let me outline a possible approach, which is very standard but I don't know a reference for this particular problem. Start with any doubly periodic embedded planar graph $\Gamma$, that is, a graph wit …

UPD: the answer below is in fact completely wrong - it deals with counting walks $\gamma$ weighted by $\mu^{-\text{length}(\gamma)}$. It is clear that without restricting or penalizing for the lengths …

The process you are looking at is called TASEP (totally asymmetric simple exclusion process); though your initial conditions are unusual. A more conventional version of your question would be to consi …

If $\varphi(\cdot)$ is a function defined on a finite $\Omega \subset \mathbb{Z}^d$, define $$ f(x):=\mathbb{E}^x\left(\sum_{t=0}^{\tau-1}\varphi(X_t)\right), $$ where $\mathbb{E}^x$ means the expecta …

There are recent lecture notes by Bauerschmidt, Duminil-Copin, Goodman, and Slade (arXiv:1206.2092). Most of the important papers should be in the references to that lecture notes, and it depends on y …