arXiv subjects

Alperen Özdemir

Publications and source records attributed to Alperen Özdemir.

6 recordsLinked to original sources

arXiv2026

The two-sided peak polynomial

We derive a generating function identity for the joint distribution of the numbers of peaks of a permutation and its inverse, via enriched $P$-partitions. The coefficients of the corresponding peak polynomial $W_n(s,t)$ satisfy a second-order recurrence. A martingale formulation of this recurrence yields a bivariate central limit theorem, showing that the two statistics are asymptotically independent. We also give an exact closed form for their covariance, which is of order $n^{-1}$.

math.CO↗

arXiv2026

First-order convergence for $321$-avoiding permutations

We say that a convergence law holds for a sequence of random combinatorial objects if, for any first-order sentence $φ$, the proportion of objects satisfying $φ$ converges to a limiting value as the size of the objects tends to infinity. In this paper, we show that the convergence law holds for random $321$-avoiding permutations, settling an open problem posed in Albert, Bouvel, Féray, and Noy (2024). Our proof relies on an infinite-dimensional version of the Perron-Frobenius theorem.

math.PR↗

arXiv2025

The $k$-Plancherel measure and a Finite Markov Chain

Let $\mathcal{P}_k(n)$ denote the set of partitions of $n$ whose largest part is bounded by $k,$ which are in well-known bijection with $(k+1)$-cores $\mathcal{C}_k$. We study a growth process on $\mathcal{C}_k$, whose stationary distribution is the $k$-Plancherel measure, which is a natural extension of the Plancherel measure in the context of $k$-Schur functions. When $k\to\infty$ it converges to the Plancherel measure for partitions, a limit studied first by Vershik-Kerov. However, when $k$ is fixed and $n\to \infty$, we conjecture that it converges to a shape close to the limit shape from the uniform growth of partitions, as studied by Rost. We show that the limiting behavior, for fixed $k$, is governed by a finite Markov chain with $k!$ states over a subset of the $k$-bounded partitions or equivalently as a TASEP over cyclic permutations of length $k+1$. This paper initiates the study of these processes, state some theorems and several intriguing conjectures found by computations of the finite Markov chain.

math.CO↗

arXiv2024

A logical limit law for the sequential model of preferential attachment graphs

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 Erdös-Renyi graphs given that the connection rate has an irrational exponent. We show a limit law for preferential attachment graphs which admit a Pólya urn representation. The two extreme cases of the parametric model, the uniform attachment graph and the sequential Barabási-Albert model, are covered separately as they exhibit qualitative differences regarding the distribution of cycles of bounded length in the graph.

math.PR↗

arXiv2022

Mixing time bounds for edge flipping on regular graphs

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

math.PR↗

arXiv2022

Fluctuation bounds for first-passage percolation on the square, tube, and torus

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 $n^{1/8-ε}$ for any $ε>0$ 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.

math.PR↗