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For a sequence of random graphs, the limit law we refer to is the existence of a limiting probability of any graph property that can be expressed in terms of predicate logic. A zero-one limit law is shown by Shelah and Spencer for Erd\"{o}s-Renyi graphs given that the connection rate has an irrational exponent. We show a limit law for preferential attachment graphs which admit a P\'{o}lya urn representation. The two extreme cases of the parametric model, the uniform attachment graph and the sequential Barab\'{a}si-Albert model, are covered separately as they exhibit qualitative differences regarding the distribution of cycles of bounded length in the graph., Comment: 42 pages

We say that a convergence law holds for a sequence of random combinatorial objects if for any first-order sentence $\varphi,$ the density of the objects that satisfy $\varphi$ converges to a limiting value. We show the convergence law for random $321$-avoiding permutations, which was left as an open problem in ABFN22. Our proof uses an infinite-dimensional version of the Perron-Frobenius theorem., Comment: 23 pages, 4 figures

The edge flipping is a non-reversible Markov chain on a given connected graph, which is defined by Chung and Graham. In the same paper, its eigenvalues and stationary distributions for some classes of graphs are identified. We further study its spectral properties to show a lower bound for the rate of convergence in the case of regular graphs. Moreover, we show that a cutoff occurs at 1/4 n log n for the edge flipping on the complete graph by a coupling argument, Comment: This manuscript has been withdrawn as a duplicate. The article is available at: arXiv:2201.03315v2

In first-passage percolation, one assigns i.i.d. nonnegative weights $(t_e)$ to the edges of $\mathbb{Z}^d$ and studies the induced distance (passage time) $T(x,y)$ between vertices $x$ and $y$. It is known that for $d=2$, the fluctuations of $T(x,y)$ are at least order $\sqrt{\log |x-y|}$ under mild assumptions on $t_e$. We study the question of fluctuation lower bounds for $T_n$, the minimal passage time between two opposite sides of an $n$ by $n$ square. The main result is that, under a curvature assumption, this quantity has fluctuations at least of order $n^{1/8-\epsilon}$ for any $\epsilon>0$ when the $t_e$ are exponentially distributed. As previous arguments to bound the fluctuations of $T(x,y)$ only give a constant lower bound for those of $T_n$ (even assuming curvature), a different argument, representing $T_n$ as a minimum of cylinder passage times, and deriving more detailed information about the distribution of cylinder times using the Markov property, is developed. As a corollary, we obtain the first polynomial lower bounds on higher central moments of the discrete torus passage time, under the same curvature assumption., Comment: 29 pages

This paper develops methods to study the distribution of Eulerian statistics defined by second-order recurrence relations. We define a random process to decompose the statistics over compositions of integers. It is shown that the numbers of descents in random involutions and in random derangements are asymptotically normal with rates of convergence $\mathcal{O}(n^{-1/2})$ and $\mathcal{O}(n^{-1/3})$ respectively., Comment: 28 pages, minor revisions made

This paper develops techniques to study the number of descents in random permutations via martingales. We relax an assumption in the Berry-Esseen theorem of Bolthausen (1982) to extend the theorem's scope to martingale differences of time-dependent variances. This extension leads to a new proof of the fact that the number of descents in random permutations is asymptotically normal with an error bound of order $1/\sqrt{n}.$ The same techniques are shown to be applicable to other descent and descent-related statistics as they satisfy certain recurrence relation conditions. These statistics include inversions, descents in signed permutations, descents in Stirling permutations, the length of the longest alternating subsequences, descents in matchings and two-sided Eulerian numbers., Comment: 27 pages. Clarifications in Section 5 and 7

This paper studies Markov chains on the symmetric group $S_n$ where the transition probabilities are given by the Ewens distribution with parameter $\theta>1$. The eigenvalues are identified to be proportional to the content polynomials of partitions. We show that the mixing time is bounded above by a constant depending only on the parameter if $\theta$ is fixed. However, if it agrees with the number of permuted elements ($\theta=n$), the sequence of chains has a total variation cutoff at $\frac{\log n}{\log 2}.$, Comment: 22 pages, 2 figures. A theorem is revised and its scope is extended. Typos are corrected

The purpose of this paper is to study a statistic that is used to compare the similarity between two strings, which is first introduced by Michael Steele in 1982. It was proposed as an alternative to the length of the longest common subsequences, for which the variance problem is still open. Our results include moment asymptotics and distributional asymptotics for Steele's statistic and a variation of it in random words and random permutations., Comment: 17 pages

We study the rate of convergence of the Markov chain on $S_n$ which starts with a random $(n-k)$-cycle for a fixed $k \geq 1$, followed by random transpositions. The convergence to the stationary distribution turns out to be of order $n$. We show that after $cn + \frac{\ln k}{2}n$ steps for $c>0$, the law of the Markov chain is close to the uniform distribution. The character of the defining representation is used as test function to obtain a lower bound for the total variation distance. We identify the asymptotic distribution of the test function given the law of the Markov chain for the $(n-1)$-cycle case. The upper bound relies on estimates for the difference of normalized characters., Comment: 23 pages, 3 figures; to appear in Journal of Theoretical Probability

Let $X=(X_1,\ldots,X_n)$ be a vector of i.i.d. random variables where $X_i$'s take values over $\mathbb{N}$. The purpose of this paper is to study the number of weakly increasing subsequences of $X$ of a given length $k$, and the number of all weakly increasing subsequences of $X$. For the former, it is shown that a central limit theorem holds. Also, the first two moments of each of those two random variables are analyzed, their asymptotics are investigated, and results are related to the case of similar statistics in uniformly random permutations. We conclude the paper with applications on a similarity measure of Steele, and on increasing subsequences of riffle shuffles., Comment: Final version