All these bijections are obtained as specializations of a general correspondence between spanning subgraphs and orientations. The definition and analysis of this correspondence are related to a characterization of the Tutte polynomial using an embedding of the graph.
In this talk, I will present a complete description of the giant component of the Erd\H{o}s-R\'enyi random graph $G(n,p)$ as soon as it emerges from the scaling window, i.e., for $p = (1+\epsilon)/n$ where $\epsilon^3 n \to \infty$ and $\epsilon=o(1)$.
Our description is particularly simple for $\epsilon = o(n^{-1/4})$, where the giant component $C_1$ is contiguous with the following model (i.e., every graph property that holds with high probability for this model also holds w.h.p. for $C_1$). Let $Z$ be normal with mean $\frac23 \epsilon^3 n$ and variance $\epsilon^3 n$, and let $K$ be a random 3-regular graph on $2[Z]$ vertices. Replace each edge of $K$ by a path, where the path lengths are i.i.d. geometric with mean $1/\epsilon$. Finally, attach an independent Poisson($1-\epsilon$)-Galton-Watson tree to each vertex.
A similar picture is obtained for larger $\epsilon=o(1)$, in which case the random 3-regular graph is replaced by a random graph with $N_k$ vertices of degree $k$ for $k\geq 3$, where $N_k$ has mean and variance of order $\epsilon^k n$.
Based on this description, we show that for any $\epsilon=o(1)$ with $\epsilon^3n\to\infty$, the diameter of $C_1$ is w.h.p. asymptotic to $D (\epsilon,n)=(3/\epsilon)\log(\epsilon^3 n)$. Furthermore, we prove that the mixing time for the random walk on $C_1$ is w.h.p. of order $\epsilon^{-3}\log^2 (\epsilon^3 n)$. Based on joint work with J.H. Kim, E. Lubetzky and Y. Peres.
We will show an elementary way of dealing with the minimizer of the logarithmic energy with external fields. This is based on manipulations of Chebyshev polynomials and combinatorial identities which give a nice new formula for the minimum of the energy. This indicates why the analyticity claim of the planar limit holds true.
We also were able to compute some of the various planar limit in closed form.
This is joint work with Stavros Garoufalidis and Marcos Marinio.