The Hammersley problem asks for the maximal number of points in a monotonous path through a Poisson point process. It is exactly solvable and notoriously known to belong to the KPZ universality class, with a cube-root scaling for the fluctuations.Here we introduce and analyze a variant in which we impose a Lipschitz condition on paths. Thanks to a coupling with the classical Hammersley problem we observe that this variant is also exactly solvable. It allows us to derive first and second orders asymptotics. It turns out that the cube-root scaling only holds for certain choices of the Lipschitz constants.
This is a commentary on the article: David Aldous and Persi Diaconis, Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem, Bull. Amer. Math. Soc. 36 (1999), no. 4, 413-432.
The longest increasing subsequence of a random walk with mean zero and finite variance is known to be $n^{1/2 + o(1)}$. We show that this is not universal for symmetric random walks. In particular, the symmetric Ultra-fat tailed random walk has a longest increasing subsequence that is asymptotically at least $n^{0.690}$ and at most $n^{0.815}$. An exponent strictly greater than $1/2$ is also shown for the symmetric stable-$α$ distribution when $α$ is sufficiently small.
Let $X_1, X_2, ..., X_n, ... $ be a sequence of iid random variables with values in a finite alphabet $\{1,...,m\}$. Let $LI_n$ be the length of the longest increasing subsequence of $X_1, X_2, ..., X_n.$ We express the limiting distribution of $LI_n$ as functionals of $m$ and $(m-1)$-dimensional Brownian motions. These expressions are then related to similar functionals appearing in queueing theory, allowing us to further establish asymptotic behaviors as $m$ grows. The finite alphabet results are then used to treat the countable (infinite) alphabet.
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.
Let $X=(X_i)_{i\ge 1}$ and $Y=(Y_i)_{i\ge 1}$ be two sequences of independent and identically distributed (iid) random variables taking their values, uniformly, in a common totally ordered finite alphabet. Let LCI$_n$ be the length of the longest common and (weakly) increasing subsequence of $X_1\cdots X_n$ and $Y_1\cdots Y_n$. As $n$ grows without bound, and when properly centered and normalized, LCI$_n$ is shown to converge, in distribution, towards a Brownian functional that we identify.
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.
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.
For two-dimensional last-passage time models of weakly increasing paths, interesting scaling limits have been proved for points close the axis (the hard edge). For strictly increasing paths of Bernoulli($p$) marked sites, the relevant boundary is the line $y=px$. We call this the soft edge to contrast it with the hard edge. We prove laws of large numbers for the maximal cardinality of a strictly increasing path in the rectangle $[\fl{p^{-1}n -xn^a}]\times[n]$ as the parameters $a$ and $x$ vary. The results change qualitatively as $a$ passes through the value 1/2.
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…
We study the fluctuations, in the large deviations regime, of the longest increasing subsequence of a random i.i.d. sample on the unit square. In particular, our results yield the precise upper and lower exponential tails for the length of the longest increasing subsequence of a random permutation.
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.
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…
A Poisson point process of unit intensity is placed in the square $[0,n]^2$. An increasing path is a curve connecting $(0,0)$ with $(n,n)$ which is non-decreasing in each coordinate. Its length is the number of points of the Poisson process which it passes through. Baik, Deift and Johansson proved that the maximal length of an increasing path has expectation $2n-n^{1/3}(c_1+o(1))$, variance $n^{2/3}(c_2+o(1))$ and that it converges to the Tracy-Widom distribution after suitable scaling. Johansson further showed that all maximal paths have a displacement of $n^{\frac23+o(1)}$ from the diagonal …
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.
In this paper, we study the reflected BSDE with one continuous barrier, under the monotonicity and general increasing condition on $y$ and non Lipschitz condition on $z$. We prove the existence and uniqueness of the solution to these equation by approximation method.
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…
Consider a rooted N-ary tree. For every vertex of this tree, we atttach an i.i.d. Bernoulli random variable. A path is called open if all the random variables that are assigned on the path are 1. We consider limiting behaviors for the number of open paths from the root to leaves and the longest open path. In addition, when all fitness values are i.i.d. continuous random variables, some asymptotic properties of the longest increasing path are proved.
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…
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.
We consider a variant of the continuous and discrete Ulam-Hammersley problems: we study the maximal length of an increasing path through a Poisson point process (or a Bernoulli point process) with the restriction that there must be minimal gaps between abscissae and ordinates of successive points of the path.For both cases (continuous and discrete) our approach rely on couplings with well-studied models: respectively the classical Ulam-Hammersley problem and last-passage percolation with geometric weights. Thanks to these couplings we obtain explicit limiting shapes in both settings.We also es…
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 $\R$.
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…
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. …
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.
We study higher order convexity properties of random point sets in the unit square. Given $n$ uniform i.i.d random points, we derive asymptotic estimates for the maximal number of them which are in $k$-monotone position, subject to mild boundary conditions. Besides determining the order of magnitude of the expectation, we also prove strong concentration estimates. We provide a general framework that includes the previously studied cases of $k=1$ (longest increasing sequences) and $k=2$ (longest convex chains).
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.
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…
We prove the existence and uniqueness of probabilistically strong solutions to stochastic porous media equations driven by time-dependent multiplicative noise on a general measure space $(E, \mathscr{B}(E), μ)$, and the Laplacian replaced by a self-adjoint operator $L$. In the case of Lipschitz nonlinearities $Ψ$, we in particular generalize previous results for open $E\subset \mathbb{R}^d$ and $L\!\!=$Laplacian to fractional Laplacians. We also generalize known results on general measure spaces, where we succeeded in dropping the transience assumption on $L$, in extending the set of allowed i…
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…
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…
Consider the graph induced by $\mathbb{Z}^d$, equipped with uniformly elliptic random conductances. At time $0$, place a Poisson point process of particles on $\mathbb{Z}^d$ and let them perform independent simple random walks. Tessellate the graph into cubes indexed by $i\in\mathbb{Z}^d$ and tessellate time into intervals indexed by $τ$. Given a local event $E(i,τ)$ that depends only on the particles inside the space time region given by the cube $i$ and the time interval $τ$, we prove the existence of a Lipschitz connected surface of cells $(i,τ)$ that separates the origin from infinity on w…
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…
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…
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…
Given a graph $G$ and a bijection $f : E(G)\rightarrow \{1, 2, \ldots,e(G)\}$, we say that a trail/path in $G$ is $f$-\emph{increasing} if the labels of consecutive edges of this trail/path form an increasing sequence. More than 40 years ago Chvátal and Komlós raised the question of providing the worst-case estimates of the length of the longest increasing trail/path over all edge orderings of $K_n$. The case of a trail was resolved by Graham and Kleitman, who proved that the answer is $n-1$, and the case of a path is still widely open. Recently Lavrov and Loh proposed to study the average cas…
We develop a new multi-scale framework flexible enough to solve a number of problems involving embedding random sequences into random sequences. Grimmett, Liggett and Richthammer asked whether there exists an increasing M-Lipschitz embedding from one i.i.d. Bernoulli sequences into an independent copy with positive probability. We give a positive answer for large enough M. A closely related problem is to show that two independent Poisson processes on R are roughly isometric (or quasi-isometric). Our approach also applies in this case answering a conjecture of Szegedy and of Peled. Our theorem …
This paper is concerned with certain connections between the ensemble of n x n unitary matrices -- specifically the characteristic function of the random variable tr(U) -- and combinatorics -- specifically Ulam's problem concerning the distribution of the length of the longest increasing subsequence in permutation groups -- and the appearance of Painleve functions in the answers to apparently unrelated questions. Among the results is a representation in terms of a Painleve V function for the characteristic function of tr(U) and (using recent results of Baik, Deift and Johansson) an expression …
We construct a stationary random tree, embedded in the upper half plane, with prescribed offspring distribution and whose vertices are the atoms of a unit Poisson point process. This process which we call Hammersley's tree process extends the usual Hammersley's line process. Just as Hammersley's process is related to the problem of the longest increasing subsequence, this model also has a combinatorial interpretation: it counts the number of heaps (i.e. increasing trees) required to store a random permutation. This problem was initially considered by Byers et. al (2011) and Istrate and Bonchis…
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…
In this note we consider the stability of posterior measures occuring in Bayesian inference w.r.t. perturbations of the prior measure and the log-likelihood function. This extends the well-posedness analysis of Bayesian inverse problems. In particular, we prove a general local Lipschitz continuous dependence of the posterior on the prior and the log-likelihood w.r.t. various common distances of probability measures. These include the total variation, Hellinger, and Wasserstein distance and the Kullback-Leibler divergence. We only assume the boundedness of the likelihoods and measure their pert…
