On increasing subsequences of iid samples

Closeness to the Diagonal for Longest Common Subsequences in Random Words

The nature of the alignment with gaps corresponding to a longest common subsequence (LCS) of two independent iid random sequences drawn from a finite alphabet is investigated. It is shown that such an optimal alignment typically matches pieces of similar short-length. This is of importance in understanding the structure of optimal alignments of two sequences. Moreover, it is also shown that any property, common to two subsequences, typically holds in most parts of the optimal alignment whenever this same property holds, with high probability, for strings of similar short-length. Our results sh…

Diffusion Approximations in the Online Increasing Subsequence Problem

The online increasing subsequence problem is a stochastic optimisation task with the objective to maximise the expected length of subsequence chosen from a random series by means of a nonanticipating decision strategy. We study the structure of optimal and near-optimal subsequences in a standardised planar Poisson framework. Following a long-standing suggestion by Bruss and Delbaen (Stoch. Proc. Appl. 114, 2004), we prove a joint functional limit theorem for the transversal fluctuations about the diagonal of the running maximum and the length processes. The limit is identified explicitly with …

GUE minors, maximal Brownian functionals and longest increasing subsequences

We present equalities in law between the spectra of the minors of a GUE matrix and some maximal functionals of independent Brownian motions. In turn, these results allow to recover the limiting shape (properly centered and scaled) of the RSK Young diagrams associated with a random word as a function of the spectra of these minors. Since the length of the top row of the diagrams is the length of the longest increasing subsequence of the random word, the corresponding limiting law also follows.

Asymptotics for the Length of the Longest Increasing Subsequence of Binary Markov Random Word

Let $(X_n)_{n\ge 0}$ be an irreducible, aperiodic, and homogeneous binary Markov chain and let $LI_n$ be the length of the longest (weakly) increasing subsequence of $(X_k)_{1\le k \le n}$. Using combinatorial constructions and weak invariance principles, we present elementary arguments leading to a new proof that (after proper centering and scaling) the limiting law of $LI_n$ is the maximal eigenvalue of a $2 \times 2$ Gaussian random matrix. In fact, the limiting shape of the RSK Young diagrams associated with the binary Markov random word is the spectrum of this random matrix.

On the limiting law of the length of the longest common and increasing subsequences in random words with arbitrary distributions

Let $(X_k)_{k\geq 1}$ and $(Y_k)_{k\geq 1}$ be two independent sequences of i.i.d. random variables, with values in a finite and totally ordered alphabet $\mathcal{A}_m:=\{1,\dots,m\}$, and having respective probability mass function $p^X_1,\dots,p^X_m$ and $p^Y_1,\dots,p^Y_m$. Let $LCI_n$ be the length of the longest common and weakly increasing subsequences in $(X_1,...,X_n)$ and $(Y_1,...,Y_n)$. Once properly centered and normalized, $LCI_n$ is shown to have a limiting distribution which is expressed as a functional of two independent multidimensional Brownian motions.

GL(n,q) and Increasing Subsequences in Nonuniform Random Permutations

Connections between longest increasing subsequences in random permutations and eigenvalues of random matrices with complex entries have been intensely studied. This note applies properties of random elements of the finite general linear group to obtain results about the longest increasing subsequence in non- uniform random permutations.

Correlation functions for symmetrized increasing subsequences

We show that the correlation functions associated to symmetrized increasing subsequence problems can be expressed as pfaffians of certain antisymmetric matrix kernels, thus generalizing the result of math.RT/9907127 for the unsymmetrized case.

Sparse Long Blocks and the Micro-Structure of the Longest Common Subsequences

Consider two random strings having the same length and generated by an iid sequence taking its values uniformly in a fixed finite alphabet. Artificially place a long constant block into one of the strings, where a constant block is a contiguous substring consisting only of one type of symbol. The long block replaces a segment of equal size and its length is smaller than the length of the strings, but larger than its square-root. We show that for sufficiently long strings the optimal alignment corresponding to a Longest Common Subsequence (LCS) treats the inserted block very differently dependi…

Increasing subsequences of random walks

Given a sequence of $n$ real numbers $\{S_i\}_{i\leq n}$, we consider the longest weakly increasing subsequence, namely $i_1<i_2<\dots <i_L$ with $S_{i_k} \leq S_{i_{k+1}}$ and $L$ maximal. When the elements $S_i$ are i.i.d. uniform random variables, Vershik and Kerov, and Logan and Shepp proved that $\mathbb{E} L=(2+o(1)) \sqrt{n}$. We consider the case when $\{S_i\}_{i\leq n}$ is a random walk on $\mathbb{R}$ with increments of mean zero and finite (positive) variance. In this case, it is well known (e.g., using record times) that the length of the longest increasing subsequence satisfies $\…

具有原子的分布中最长递增子序列，以及非均匀的Hammersley过程

Longest increasing subsequences for distributions with atoms, and an inhomogeneous Hammersley process

本文研究了具有原子的分布中最长递增子序列的渐进行为，通过变分问题和非均匀Hammersley过程的耦合，揭示了离散情况下增长率的广泛范围。

Increasing and Decreasing Subsequences of Permutations and Their Variants

We survey the theory of increasing and decreasing subsequences of permutations. Enumeration problems in this area are closely related to the RSK algorithm. The asymptotic behavior of the expected value of the length is(w) of the longest increasing subsequence of a permutation w of 1,2,...,n was obtained by Vershik-Kerov and (almost) by Logan-Shepp. The entire limiting distribution of is(w) was then determined by Baik, Deift, and Johansson. These techniques can be applied to other classes of permutations, such as involutions, and are related to the distribution of eigenvalues of elements of the…

Longest increasing subsequences of random colored permutations

We compute the limit distribution for (centered and scaled) length of the longest increasing subsequence of random colored permutations. The limit distribution function is a power of that for usual random permutations computed recently by Baik, Deift, and Johansson (math.CO/9810105). In two--colored case our method provides a different proof of a similar result by Tracy and Widom about longest increasing subsequences of signed permutations (math.CO/9811154). Our main idea is to reduce the `colored' problem to the case of usual random permutations using certain combinatorial results and element…

Asymptotics and Renewal Approximation in the Online Selection of Increasing Subsequence

We revisit the problem of maximising the expected length of increasing subsequence that can be selected from a marked Poisson process by an online strategy. Resorting to a natural size variable, the problem is represented in terms of a controlled partially deterministic Markov process with decreasing paths. Refining known estimates we obtain fairly complete asymptotic expansions for the moments, and using a renewal approximation give a novel proof of the central limit theorem for the length of selected subsequence under the optimal strategy.

A mean identity for longest increasing subsequence problems

We show that a wide variety of generalized increasing subsequence problems admit a one parameter family of extensions for which we can exactly compute the mean length of the longest increasing subsequence. By the nature of the extension, this gives upper bounds on the mean in the unextended model, which turn out to be asymptotically tight for all of the models that have so far been analyzed. A heuristic analysis based on this fact gives not just the asymptotic mean but also the asymptotic scale factor, again agreeing with all known cases.

The Preisach graph and longest increasing subsequences

The Preisach graph is a directed graph associated with a permutation $ρ\in{\cal S}_N$. We give an explicit bijection between its vertices and increasing subsequences of $ρ$ with the property that the length of a subsequence equals to the degree of nesting of the corresponding vertex inside a hierarchy of cycles and sub-cycles of the graph. As a consequence, the nesting degree of the Preisach graph equals the length of the longest increasing subsequence.

Inversions and Longest Increasing Subsequence for $k$-Card-Minimum Random Permutations

A random $n$-permutation may be generated by sequentially removing random cards $C_1,...,C_n$ from an $n$-card deck $D = \{1,...,n\}$. The permutation $σ$ is simply the sequence of cards in the order they are removed. This permutation is itself uniformly random, as long as each random card $C_t$ is drawn uniformly from the remaining set at time $t$. We consider, here, a variant of this simple procedure in which one is given a choice between $k$ random cards from the remaining set at each step, and selects the lowest numbered of these for removal. This induces a bias towards selecting lower num…

Applications of Symmetric Functions to Cycle and Increasing Subsequence Structure after Shuffles (Part 2)

Using the Berele/Remmel/Kerov/Vershik variation of the Robinson-Schensted-Knuth correspondence, we study the cycle and increasing subsequence structure after various methods of shuffling. One consequence is a cycle index for shuffles like: cut a deck into roughly two equal piles, thoroughly mix the first pile and then riffle it with the second pile. Conclusions are drawn concerning the distribution of fixed points and the asymptotic distribution of cycle structure. An upper bound on the convergence rate is given. Connections are made with extended Schur functions and with work of Baik and Rain…

Algebraic aspects of increasing subsequences

We present a number of results relating partial Cauchy-Littlewood sums, integrals over the compact classical groups, and increasing subsequences of permutations. These include: integral formulae for the distribution of the longest increasing subsequence of a random involution with constrained number of fixed points; new formulae for partial Cauchy-Littlewood sums, as well as new proofs of old formulae; relations of these expressions to orthogonal polynomials on the unit circle; and explicit bases for invariant spaces of the classical groups, together with appropriate generalizations of the str…

Quickest Online Selection of an Increasing Subsequence of Specified Size

Given a sequence of independent random variables with a common continuous distribution, we consider the online decision problem where one seeks to minimize the expected value of the time that is needed to complete the selection of a monotone increasing subsequence of a prespecified length $n$. This problem is dual to some online decision problems that have been considered earlier, and this dual problem has some notable advantages. In particular, the recursions and equations of optimality lead with relative ease to asymptotic formulas for mean and variance of the minimal selection time.

Law of large numbers for increasing subsequences of random permutations and an approximation result for the uniform measure

Let the random variable $Z_{n,k}$ denote the number of increasing subsequences of length $k$ in a random permutation from $S_n$, the symmetric group of permutations of $\{1,...,n\}$. We show that $Var(Z_{n,k_n})=o((EZ_{n,k_n})^2)$ as $ n\to\infty$ if and only if $k_n=o(n^\frac25)$. In particular then, the weak law of large numbers holds for $Z_{n,k_n}$ if $k_n=o(n^\frac25)$. We also show the following approximation result for the uniform measure $U_n$ on $S_n$. Define the probability measure $μ_{n;k_n}$ on $S_n$ as follows: Consider $n$ cards, numbered from 1 to $n$, and laid out on a table fr…

关于所有重数均相等的词中的最长递增子序列

On longest increasing subsequences in words in which all multiplicities are equal

本文针对所有重数相等的多重集排列推广了Gessel的Bessel行列式公式，提出一个表示为Gessel型Toeplitz行列式与对称群轮换指标多项式乘积多重积分的生成函数，用于统计不含特定长度递增子序列的词。

The length of the longest increasing subsequence of Mallows permutation models with $L^1$ and $L^2$ distances

Introduced by Mallows in statistical ranking theory, Mallows permutation model is a class of non-uniform probability measures on the symmetric group $S_n$ that depend on a distance metric $d(σ,τ)$ on $S_n$ and a scale parameter $β$. Taking the distance metric to be the $L^1$ and $L^2$ distances--which are respectively known as Spearman's footrule and Spearman's rank correlation in the statistics literature--leads to Mallows permutation models with $L^1$ and $L^2$ distances. In this paper, we study the length of the longest increasing subsequence of random permutations drawn from Mallows permut…

Longest increasing subsequences, Plancherel-type measure and the Hecke insertion algorithm

We define and study the Plancherel-Hecke probability measure on Young diagrams; the Hecke algorithm of [Buch-Kresch-Shimozono-Tamvakis-Yong '06] is interpreted as a polynomial-time exact sampling algorithm for this measure. Using the results of [Thomas-Yong '07] on jeu de taquin for increasing tableaux, a symmetry property of the Hecke algorithm is proved, in terms of longest strictly increasing/decreasing subsequences of words. This parallels classical theorems of [Schensted '61] and of [Knuth '70], respectively, on the Schensted and Robinson-Schensted-Knuth algorithms. We investigate, and co…

On the rate of approximation in finite-alphabet longest increasing subsequence problems

The rate of convergence of the distribution of the length of the longest increasing subsequence, toward the maximal eigenvalue of certain matrix ensembles, is investigated. For finite-alphabet uniform and nonuniform i.i.d. sources, a rate of $\log n/\sqrt{n}$ is obtained. The uniform binary case is further explored, and an improved $1/\sqrt{n}$ rate obtained.

Lengths of Monotone Subsequences in a Mallows Permutation

We study the length of the longest increasing and longest decreasing subsequences of random permutations drawn from the Mallows measure. Under this measure, the probability of a permutation pi in S_n is proportional to q^{inv(pi)} where q is a real parameter and inv(pi) is the number of inversions in pi. The case q=1 corresponds to uniformly random permutations. The Mallows measure was introduced by Mallows in connection with ranking problems in statistics. We determine the typical order of magnitude of the lengths of the longest increasing and decreasing subsequences, as well as large deviati…

The Length of the Longest Increasing Subsequence of a Random Mallows Permutation

The Mallows measure on the symmetric group $S_n$ is the probability measure such that each permutation has probability proportional to $q$ raised to the power of the number of inversions, where $q$ is a positive parameter and the number of inversions of $π$ is equal to the number of pairs $i<j$ such that $π_i > π_j$. We prove a weak law of large numbers for the length of the longest increasing subsequence for Mallows distributed random permutations, in the limit that $n$ tends to infinity and $q$ tends to 1 in such a way that $n(1-q)$ has a limit in $\R$.

Continuously Increasing Subsequences of Random Multiset Permutations

For a word $π$ and integer $i$, we define $L^i(π)$ to be the length of the longest subsequence of the form $i(i+1)\cdots j$, and we let $L(π):=\max_i L^i(π)$. In this paper we estimate the expected values of $L^1(π)$ and $L(π)$ when $π$ is chosen uniformly at random from all words which use each of the first $n$ integers exactly $m$ times. We show that $\mathbb{E}[L^1(π)]\sim m$ if $n$ is sufficiently larger in terms of $m$ as $m$ tends towards infinity, confirming a conjecture of Diaconis, Graham, He, and Spiro. We also show that $\mathbb{E}[L(π)]$ is asymptotic to the inverse gamma function …

The scaling limit of the longest increasing subsequence

We provide a framework for proving convergence to the directed landscape, the central object in the Kardar-Parisi-Zhang universality class. For last passage models, we show that compact convergence to the Airy line ensemble implies convergence to the Airy sheet. In i.i.d. environments, we show that Airy sheet convergence implies convergence of distances and geodesics to their counterparts in the directed landscape. Our results imply convergence of classical last passage models and interacting particle systems. Our framework is built on the notion of a directed metric, a generalization of metri…

Analysis of disk scheduling, increasing subsequences and space-time geometry

We consider the problem of estimating the average tour length of the asymmetric TSP arising from the disk scheduling problem with a linear seek function and a probability distribution on the location of I/O requests. The optimal disk scheduling algorithm of Andrews, Bender and Zhang is interpreted as a simple peeling process on points in a 2 dimensional space-time w.r.t the causal structure. The patience sorting algorithm for finding the longest increasing subsequence in a permutation can be given a similar interpretation. Using this interpretation we show that the optimal tour length is the l…

Empirical scaling of the length of the longest increasing subsequences of random walks

We provide Monte Carlo estimates of the scaling of the length $L_{n}$ of the longest increasing subsequences of $n$-steps random walks for several different distributions of step lengths, short and heavy-tailed. Our simulations indicate that, barring possible logarithmic corrections, $L_{n} \sim n^θ$ with the leading scaling exponent $0.60 \lesssim θ\lesssim 0.69$ for the heavy-tailed distributions of step lengths examined, with values increasing as the distribution becomes more heavy-tailed, and $θ\simeq 0.57$ for distributions of finite variance, irrespective of the particular distribution. …

Linear-sized independent sets in random cographs and increasing subsequences in separable permutations

This paper is interested in independent sets (or equivalently, cliques) in uniform random cographs. We also study their permutation analogs, namely, increasing subsequences in uniform random separable permutations. First, we prove that, with high probability as $n$ gets large, the largest independent set in a uniform random cograph with $n$ vertices has size $o(n)$. This answers a question of Kang, McDiarmid, Reed and Scott. Using the connection between graphs and permutations via inversion graphs, we also give a similar result for the longest increasing subsequence in separable permutations. …

二进型混沌轨道的最长递增子序列

Longest increasing subsequences of dyadic-type chaotic orbits

该研究针对二进型混沌区间映射生成的序列，计算其最长递增子序列长度的渐近行为，发现其与均匀随机排列经典问题的主导渐近结果一致。

An adaptive $O(\log n)$-optimal policy for the online selection of a monotone subsequence from a random sample

Given a sequence of $n$ independent random variables with common continuous distribution, we propose a simple adaptive online policy that selects a monotone increasing subsequence. We show that the expected number of monotone increasing selections made by such a policy is within $O(\log n)$ of optimal. Our construction provides a direct and natural way for proving the $O(\log n)$-optimality gap. An earlier proof of the same result made crucial use of a key inequality of Bruss and Delbaen (2001) and of de-Poissonization.

Limit Theorems for Longest Monotone Subsequences in Random Mallows Permutations

We study the lengths of monotone subsequences for permutations drawn from the Mallows measure. The Mallows measure was introduced by Mallows in connection with ranking problems in statistics. Under this measure, the probability of a permutation $π$ is proportional to $q^{{\rm inv}(π)}$ where $q$ is a positive parameter and ${\rm inv}(π)$ is the number of inversions in $π$. In our main result we show that when $0<q<1$, the limiting distribution of the longest increasing subsequence (LIS) is Gaussian, answering an open question in [Bhatnagar and Peled, PTRF, 2015]. This is in contrast to the cas…

具有大递增子序列的局部均匀随机排列

Locally uniform random permutations with large increasing subsequences

本文研究随机采样点中递增子集的最大大小，给出密度的两个充分条件，使增长速率可达到大于根号N的任意幂次，方法为网格切片分析。

When the law of large numbers fails for increasing subsequences of random permutations

Let the random variable $Z_{n,k}$ denote the number of increasing subsequences of length $k$ in a random permutation from $S_n$, the symmetric group of permutations of $\{1,...,n\}$. In a recent paper [Random Structures Algorithms 29 (2006) 277--295] we showed that the weak law of large numbers holds for $Z_{n,k_n}$ if $k_n=o(n^{2/5})$; that is, \[\lim_{n\to\infty}\frac{Z_{n,k_n}}{EZ_{n,k_n}}=1\qquad in probability.\] The method of proof employed there used the second moment method and demonstrated that this method cannot work if the condition $k_n=o(n^{2/5})$ does not hold. It follows from re…

On the rate of convergence of the shape of Young diagrams associated with random words

We revisit, beyond the uniform case, some aspects of the convergence of the cumulative shape of the RSK Young diagrams associated with random words, obtaining rates of convergence in Kolmogorov's distance. Since the length of the top row of the diagrams is the length of the longest increasing subsequences of the word, a corresponding rate result follows. This is then extended to the length of the longest common and increasing subsequences in two or more random words.

Transversal fluctuations for increasing subsequences on the plane

Consider a realization of a Poisson process in R^2 with intensity 1 and take a maximal up/right path from the origin to (N,N) consisting of line segments between the points, where maximal means that it contains as many points as possible. The number of points in such a path has fluctuations of order N^chi, where chi=1/3 by a result of Baik-Deift-Johansson. Here we show that typical deviations of a maximal path from the diagonal x=y is of order N^xi with xi=2/3. This is consistent with the scaling identity chi=2xi-1, which is believed to hold in many random growth models.

避免长递减序列的排列的缩放极限

Scaling limits of permutations avoiding long decreasing sequences

研究避免模式(d+1)d...21的排列的缩放极限，证明递增子序列经缩放和中心化后收敛于无迹d×d Hermitian布朗桥的特征值过程。

A numerical investigation into the scaling behavior of the longest increasing subsequences of the symmetric ultra-fat tailed random walk

The longest increasing subsequence (LIS) of a sequence of correlated random variables is a basic quantity with potential applications that has started to receive proper attention only recently. Here we investigate the behavior of the length of the LIS of the so-called symmetric ultra-fat tailed random walk, introduced earlier in an abstract setting in the mathematical literature. After explicit constructing the ultra-fat tailed random walk, we found numerically that the expected length $L_{n}$ of its LIS scales with the length $n$ of the walk like $\langle L_{n} \rangle \sim n^{0.716}$, indica…

随机排列中线性大小的递增子序列与permutons的Robinson-Schensted表

Increasing subsequences of linear size in random permutations and the Robinson-Schensted tableaux of permutons

本文扩展Robinson-Schensted对应到permutons，证明其RS-表非平凡时，采样排列的递增子序列呈线性增长，并给出大偏差结果及偏微分方程。

Asymptotic behavior of the length of the longest increasing subsequences of random walks

We numerically estimate the leading asymptotic behavior of the length $L_{n}$ of the longest increasing subsequence of random walks with step increments following Student's $t$-distribution with parameter in the range $1/2 \leq ν\leq 5$. We find that the expected value $\mathbb{E}(L_{n}) \sim n^θ\ln{n}$ with $θ$ decreasing from $θ(ν=1/2) \approx 0.70$ to $θ(ν\geq 5/2) \approx 0.50$. For random walks with distribution of step increments of finite variance ($ν> 2$), this confirms previous observation of $\mathbb{E}(L_{n}) \sim \sqrt{n}\ln{n}$ to leading order. We note that this asymptotic behavi…

Permutations without long decreasing subsequences and random matrices

We study the shape of the Young diagram λassociated via the Robinson-Schensted-Knuth algorithm to a random permutation in S_n such that the length of the longest decreasing subsequence is not bigger than a fixed number d; in other words we study the restriction of the Plancherel measure to Young diagrams with at most d rows. We prove that in the limit n\to\infty the rows of λbehave like the eigenvalues of a certain random matrix (traceless Gaussian Unitary Ensemble) with d rows and columns. In particular, the length of the longest increasing subsequence of such a random permutation behaves asy…

Optimal Online Selection of a Monotone Subsequence: a Central Limit Theorem

Consider a sequence of $n$ independent random variables with a common continuous distribution $F$, and consider the task of choosing an increasing subsequence where the observations are revealed sequentially and where an observation must be accepted or rejected when it is first revealed. There is a unique selection policy $π_n^*$ that is optimal in the sense that it maximizes the expected value of $L_n(π_n^*)$, the number of selected observations. We investigate the distribution of $L_n(π_n^*)$; in particular, we obtain a central limit theorem for $L_n(π_n^*)$ and a detailed understanding of i…

Finite size corrections relating to distributions of the length of longest increasing subsequences

Considered are the large $N$, or large intensity, forms of the distribution of the length of the longest increasing subsequences for various models. Earlier work has established that after centring and scaling, the limit laws for these distributions relate to certain distribution functions at the hard edge known from random matrix theory. By analysing the hard to soft edge transition, we supplement and extend results of Baik and Jenkins for the Hammersley model and symmetrisations, which give that the leading correction is proportional to $z^{-2/3}$, where $z^2$ is the intensity of the Poisson…

Monotonous subsequences and the descent process of invariant random permutations

It is known from the work of Baik, Deift, and Johansson [1999] that we have Tracy-Widom fluctuations for the longest increasing subsequence of uniform permutations. In this paper, we prove that this result holds also in the case of the Ewens distribution and more generally for a class of random permutation with distribution invariant under conjugation. Moreover, we obtain the convergence of the first components of the associated Young tableaux to the Airy Ensemble as well as the global convergence to the Vershik-Kerov-Logan-Shepp shape. Using similar techniques, we also prove that the limiting…

Limit Theorems for the Length of the Longest Common Subsequence of Mallows Permutations

The Mallows measure is measure on permutations which was introduced by Mallows in connection with ranking problems in statistics. Under this measure, the probability of a permutation $π$ is proportional to $q^{Inv(π)}$ where $q$ is a positive parameter and $Inv(π)$ is the number of inversions in $π$. We consider the length of the longest common subsequence (LCS) of two independently permutations drawn according to $μ_{n,q}$ and $μ_{n,q'}$ for some $q,q' >0$. We show that when $0<q,q'<1$, the limiting law of the LCS is Gaussian. In the regime that $n(1-q) \to \infty$ and $n(1-q') \to \infty$ we…

Variations on Hammersley's interacting particle process

The longest increasing subsequence problem for permutations has been studied extensively in the last fifty years. The interpretation of the longest increasing subsequence as the longest 21-avoiding subsequence in the context of permutation patterns leads to many interesting research directions. We introduce and study the statistical properties of Hammersleytype interacting particle processes related to these generalizations and explore the finer structures of their distributions. We also propose three different interacting particle systems in the plane analogous to the Hammersley process in on…

On the Distributions of the Lengths of the Longest Monotone Subsequences in Random Words

We consider the distributions of the lengths of the longest weakly increasing and strongly decreasing subsequences in words of length N from an alphabet of k letters. We find Toeplitz determinant representations for the exponential generating functions (on N) of these distribution functions and show that they are expressible in terms of solutions of Painlevé V equations. We show further that in the weakly increasing case the generating function gives the distribution of the smallest eigenvalue in the k x k Laguerre random matrix ensemble and that the distribution itself has, after centering an…

布朗可分离置换子（Brownian separable permutons）的最长递增子序列

The longest increasing subsequence of Brownian separable permutons

本研究证明从布朗可分离置换子采样的置换的最长递增子序列长度除以$n^{\alpha(p)}$几乎必然收敛到非平凡随机变量，其中指数$\alpha(p)$由特定方程唯一确定，并推广至布朗cographon的最大团大小。

避免长度为三的模式的随机排列中某些重复上升/下降模式的最长子序列

Longest subsequence for certain repeated up/down patterns in random permutations avoiding a pattern of length three

本文研究避免模式132的随机排列中重复UUD与UUUD的最长子序列，证明其期望长度分别渐近于3n/7和4n/11，并推广至所有长度为三的避免模式。

最长递增子序列长度分布的渐近展开

Asymptotic expansions relating to the distribution of the length of longest increasing subsequences

本文研究随机排列中最长递增子序列长度的分布，建立了以$n^{-1/3}$为幂次的渐近展开，给出有限尺寸修正项的显式表达式，并采用解析去泊松化方法证明，同时提供了期望和方差的展开。

The speed of biased random walk among random conductances

We consider biased random walk among iid, uniformly elliptic conductances on $\mathbb{Z}^d$, and investigate the monotonicity of the velocity as a function of the bias. It is not hard to see that if the bias is large enough, the velocity is increasing as a function of the bias. Our main result is that if the disorder is small, i.e. all the conductances are close enough to each other, the velocity is always strictly increasing as a function of the bias, see Theorem 1. A crucial ingredient of the proof is a formula for the derivative of the velocity, which can be written as a covariance, see The…

记录、漂移与有偏高斯随机游走的最长递增子序列

Records, drift, and the longest increasing subsequence of biased Gaussian random walks

本文通过数值研究有偏高斯随机游走的最长递增子序列，发现当漂移为正时平均长度线性增长，系数从0增至1，且与记录骨架对齐；对称点处恢复为平方根对数尺度。