Catalan numbers arise in many enumerative contexts as the counting sequence of combinatorial structures. In this work, we consider natural Markov chains on some of the realizations of the Catalan sequence. While our main result is in deriving an $O(n^2 log n)$ bound on the mixing time in $L_2$ (and hence total variation) distance for the random transposition chain on Dyck paths, we raise several open questions, including the optimality of the above bound. The novelty in our proof is in establishing a certain negative correlation property among random bases of lattice path matroids, including the so-called Catalan matroid which can be defined using Dyck paths.
Given a capacitated undirected graph $G=(V,E)$ with a set of terminals $K subset V$, a mimicking network is a smaller graph $H=(V_H,E_H)$ that exactly preserves all the minimum cuts between the terminals. Specifically, the vertex set of the sparsifier $V_H$ contains the set of terminals $K$ and for every bipartition $U, K-U $ of the terminals $K$, the size of the minimum cut separating $U$ from $K-U$ in $G$ is exactly equal to the size of the minimum cut separating $U$ from $K-U$ in $H$. This notion of a mimicking network was introduced by Hagerup, Katajainen, Nishimura and Ragde (1995) who also exhibited a mimicking network of size $2^{2^{k}}$ for every graph with $k$ terminals. The best known lower bound on the size of a mimicking network is linear in the number of terminals. More precisely, the best known lower bound is $k+1$ for graphs with $k$ terminals (Chaudhuri et al. 2000). In this work, we improve both the upper and lower bounds reducing the doubly-exponential gap between them to a single-exponential gap. Specifically, we obtain the following upper and lower bounds on mimicking networks: 1) Given a graph $G$, we exhibit a construction of mimicking network with at most $(|K|-1)$th Dedekind number ($approx 2^{{(k-1)} choose {lfloor {{(k-1)}/2} rfloor}}$) of vertices (independent of size of $V$). Furthermore, we show that the construction is optimal among all {it restricted mimicking networks} -- a natural class of mimicking networks that are obtained by clustering vertices together. 2) There exists graphs with $k$ terminals that have no mimicking network of size smaller than $2^{frac{k-1}{2}}$. We also exhibit improved constructions of mimicking networks for trees and graphs of bounded tree-width.
We establish the existence of free energy limits for several combinatorial models on Erd{o}s-R{e}nyi graph $mathbb {G}(N,lfloor cNrfloor)$ and random $r$-regular graph $mathbb {G}(N,r)$. For a variety of models, including independent sets, MAX-CUT, coloring and K-SAT, we prove that the free energy both at a positive and zero temperature, appropriately rescaled, converges to a limit as the size of the underlying graph diverges to infinity. In the zero temperature case, this is interpreted as the existence of the scaling limit for the corresponding combinatorial optimization problem. For example, as a special case we prove that the size of a largest independent set in these graphs, normalized by the number of nodes converges to a limit w.h.p. This resolves an open problem which was proposed by Aldous (Some open problems) as one of his six favorite open problems. It was also mentioned as an open problem in several other places: Conjecture 2.20 in Wormald [In Surveys in Combinatorics, 1999 (Canterbury) (1999) 239-298 Cambridge Univ. Press]; Bollob{a}s and Riordan [Random Structures Algorithms 39 (2011) 1-38]; Janson and Thomason [Combin. Probab. Comput. 17 (2008) 259-264] and Aldous and Steele [In Probability on Discrete Structures (2004) 1-72 Springer].
