arXiv subjects

Boris Pittel

Publications and source records attributed to Boris Pittel.

At least 37 records · Page 2Linked to original sources

arXiv2018

Asymptotic joint distribution of the extremities of a random Young diagram and enumeration of graphical partitions

An integer partition of $n$ is a decreasing sequence of positive integers that add up to $[n]$. Back in $1979$ Macdonald posed a question about the limit value of the probability that two partitions chosen uniformly at random, and independently of each other, are comparable in terms of the dominance order. In $1982$ Wilf conjectured that the uniformly random partition is a size-ordered degree sequence of a simple graph with the limit probability $0$. In $1997$ we showed that in both, seemingly unrelated, cases the limit probabilities are indeed zero, but our method left open the problem of convergence rates. The main result in this paper is that each of the probabilities is $e^{-0.11\log n/\log\log n}$, at most. A key element of the argument is a local limit theorem, with convergence rate, for the joint distribution of the $[n^{1/4-\varepsilon}]$ tallest columns and the $[n^{1/4-\varepsilon}]$ longest rows of the Young diagram representing the random partition.

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arXiv2018

On sparsity of the solution to a random quadratic optimization problem

The standard quadratic optimization problem (StQP), i.e. the problem of minimizing a quadratic form $\bold x^TQ\bold x$ on the standard simplex $\{\bold x\ge\bold 0: \bold x^T\bold e=1\}$, is studied. The StQP arises in numerous applications, and it is known to be NP-hard. The first author, Peng and Zhang~\cite{int:Peng-StQP} showed that almost certainly the StQP with a large random matrix $Q=Q^T$, whose upper-triangular entries are i. i. concave-distributed, attains its minimum at a point with few positive components. In this paper we establish sparsity of the solution for a considerably broader class of the distributions, including those supported by $(-\infty,\infty)$, provided that the distribution tail is (super/sub)-exponentially narrow, and also for the matrices $Q=(M+M^T)/2$, when $M$ is not symmetric. {The likely support size in those cases is shown to be polylogarithmic in $n$, the problem dimension.} Following~\cite{int:Peng-StQP} and Chen and Peng ~\cite{ChenPeng2015}, the key ingredients are the first and second order optimality conditions, and the integral bound for the tail distribution of the solution support size. To make these results work for our goal, we obtain a series of estimates involving, in particular, the random interval partitions induced by the order statistics of the elements $Q_{i,j}$.

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arXiv2018

On random quadratic forms: supports of potential local maxima

In the late eighties John Kingman studied the problem of maxima of a quadratic form, with independent, uniformly distributed, coefficients, on a simplex of growing dimension $n$. In particular, he proved that the largest support size (cardinality) $L_n$ of a potential local maximum is, in probability, $2.49 n^{1/2}$ at most, and for a non-biological case of independent exponentials on $[0,\infty)$ he reduced the constant to $2.14$. In this paper we show that the constant $2.14$ serves a broad class of the densities on $[0,1]$, which includes a linear non-decreasing (whence uniform) density and the exponential density conditioned on $[0,1]$. We also prove a qualitatively matching lower bound: in probability, $L_n\ge 2n^{1/3}$ at least. Our argument shows also that the random counts of potential maxima supports, whose sizes range from $2$ to $\lceil 2n^{1/3}\rceil$, are asymptotic to their expected values. Finally we show that a support of a local maximum, that does not contain a support of a local equilibrium, is very unlikely to have size exceeding $2\log_2 n$.

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arXiv2017

On random exchange-stable matchings

Consider the group of $n$ men and $n$ women, each with their own preference list for a potential marriage partner. The stable marriage is a bipartite matching such that no unmatched pair (man, woman) prefer each other to their partners in the matching. Its non-bipartite version, with an even number $n$ of members, is known as the stable roommates problem. Jose Alcalde introduced an alternative notion of exchange-stable, one-sided, matching: no two members prefer each other's partners to their own partners in the matching. Katarina Cechlárová and David Manlove showed that the e-stable matching decision problem is $NP$-complete for both types of matchings. We prove that the expected number of e-stable matchings is asymptotic to $\left(\frac{πn}{2}\right)^{1/2}$ for two-sided case, and to $e^{1/2}$ for one-sided case. However, the standard deviation of this number exceeds $1.13^n$, ($1.06^n$ resp.). As an obvious byproduct, there exist instances of preference lists with at least $1.13^n$ ($1.06^n$ resp.) e-stable matchings. The probability that there is no matching which is stable and e-stable is at least $1-e^{-n^{1/6+o(1)}}$, ($1-O(2^{-n/2})$ resp.).

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arXiv2017

On random stable partitions

The stable roommates problem does not necessarily have a solution, i.e. a stable matching. We had found that, for the uniformly random instance, the expected number of solutions converges to $e^{1/2}$ as $n$, the number of members, grows, and with Rob Irving we proved that the limiting probability of solvability is $e^{1/2}/2$, at most. Stephan Mertens's extensive numerics compelled him to conjecture that this probability is of order $n^{-1/4}$. Jimmy Tan introduced a notion of a stable cyclic partition, and proved existence of such a partition for every system of members' preferences, discovering that presence of odd cycles in a stable partition is equivalent to absence of a stable matching. In this paper we show that the expected number of stable partitions with odd cycles grows as $n^{1/4}$. However the standard deviation of that number is of order $n^{3/8}\gg n^{1/4}$, too large to conclude that the odd cycles exist with high probability (whp). Still, as a byproduct, we show that whp the fraction of members with more than one stable "predecessor" is of order $n^{-1/4}$. Furthermore, whp the average rank of a predecessor in every stable partition is of order $n^{1/2}$. The likely size of the largest stable matching is $n/2-O(n^{1/4+o(1)})$, and the likely number of pairs of unmatched members blocking the optimal complete matching is $O(n^{3/4+o(1)})$.

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arXiv2017

On likely solutions of the stable matching problem with unequal numbers of men and women

Following up a recent work by Ashlagi, Kanoria and Leshno, we study a stable matching problem with unequal numbers of men and women, and independent uniform preferences. The asymptotic formulas for the expected number of stable matchings, and for the probabilities of one point--concentration for the range of husbands' total ranks and for the range of wives' total ranks are obtained.

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arXiv2016

On the cycle structure of the product of random maximal cycles

The subject of this paper is the cycle structure of the random permutation $σ$ of $[N]$, which is the product of $k$ independent random cycles of maximal length $N$. We use the character-based Fourier transform to study the number of cycles of $σ$ and also the distribution of the elements of the subset $[\ell]$ among the cycles of $σ$.

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arXiv2016

Counting strongly connected $(k_1,k_2)$-directed cores

Consider the set of all digraphs on $[N]$ with $M$ edges, whose minimum in-degree and minimum out-degree are at least $k_1$ and $k_2$ respectively. For $k:=\min\{k_1,k_2\}\ge 2$ and $M/N>\max\{k_1,k_2\}$, $M=Θ(N)$, we show that, among those digraphs, the fraction of $k$-strongly connected digraphs is $1-O\bigl(N^{-(k-1)})$. Earlier with Dan Poole we identified a sharp edge-density threshold $c^*(k_1,k_2)$ for birth of a giant $(k_1,k_2)$-core in the random digraph $D(n,m=[cn])$. Combining the claims, for $c>c^*(k_1,k_2)$ with probability $1-O\bigl(N^{-(k-1)})$ the giant $(k_1,k_2)$-core exists and is $k$-strongly connected.

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arXiv2016

Birth of a giant $(k_1,k_2)$-core in the random digraph

The $(k_1,k_2)$-core of a digraph is the largest sub-digraph with minimum in-degree and minimum out-degree at least $k_1$ and $k_2$ respectively. For $\max\{k_1, k_2\} \geq 2$, we establish existence of the threshold edge-density $c^*=c^*(k_1,k_2)$, such that the random digraph $D(n,m)$, on the vertex set $[n]$ with $m$ edges, asymptotically almost surely has a giant $(k_1,k_2)$-core if $m/n> c^*$, and has no $(k_1,k_2)$-core if $m/n<c^*$. Specifically, denoting $\text{P}(\text{Poisson}(z)\ge k)$ by $p_k(z)$, we prove that $c^*=\min\limits_{z_1,z_2}\max\left\{\tfrac{z_1}{p_{k_1}(z_1)p_{k_2-1}(z_2)}; \tfrac{z_2}{p_{k_1-1}(z_1)p_{k_2}(z_2)}\right\}$.

math.PR↗

arXiv2015

On a random search tree: asymptotic enumeration of vertices by distance from leaves

A random binary search tree grown from the uniformly random permutation of $[n]$ is studied. We analyze the exact and asymptotic counts of vertices by rank, the distance from the set of leaves. The asymptotic fraction $c_k$ of vertices of a fixed rank $k\ge 0$ is shown to decay exponentially with $k$. Notoriously hard to compute, the exact fractions $c_k$ had been determined for $k\le 3$ only. We computed $c_4$ and $c_5$ as well; both are ratios of enormous integers, denominator of $c_5$ being $274$ digits long. Prompted by the data, we proved that, in sharp contrast, the largest prime divisor of $c_k$'s denominator is $2^{k+1}+1$ at most. We conjecture that, in fact, the prime divisors of every denominator for $k>1$ form a single interval, from $2$ to the largest prime not exceeding $2^{k+1}+1$.

math.CO↗

arXiv2015

Asymptotic distribution of the numbers of vertices and arcs of the giant strong component in sparse random digraphs

Two models of a random digraph on $n$ vertices, $D(n,\text{Prob}(\text{arc})=p)$ and $D(n,\text{number of arcs}=m)$ are studied. In 1990, Karp for $D(n,p)$ and independently T. Łuczak for $D(n,m=cn)$ proved that for $c>1$, with probability tending to 1, there is an unique strong component of size of order $n$. Karp showed, in fact, that the giant component has likely size asymptotic to $nθ^2$, where $θ=θ(c)$ is the unique positive root of $1-θ=e^{-c θ}$. In this paper we prove that, for both random digraphs, the joint distribution of the number of vertices and number of arcs in the giant strong component is asymptotically Gaussian with the same mean vector $n\boldsymbolμ(c)$, $\boldsymbolμ(c):=(θ^2, cθ^2)$ and two distinct $2\times 2$ covariance matrices, $n\mathbf{B}(c)$ and $n[\mathbf{B}(c)+c (\boldsymbolμ'(c))^T (\boldsymbolμ'(c)))]$. To this end, we introduce and analyze a randomized deletion process which determines the directed $(1,1)$-core, the maximal digraph with minimum in-degree and out-degree at least 1. This $(1,1)$-core contains all non-trivial strong components. However, we show that the likely numbers of peripheral vertices and arcs in the $(1,1)$-core, those outside the largest strong component, are of log-polynomial order, thus dwarfed by anticipated fluctuations, on the scale of $n^{1/2}$, of the giant component parameters. By approximating the likely realization of the deletion algorithm with a deterministic trajectory, we obtain our main result via exponential supermartingales and Fourier-based techniques.

math.PR↗

arXiv2015

Another proof of Harer-Zagier formula

For a regular $2n$-gon there are $(2n-1)!!$ ways to match and glue the $2n$ sides. The Harer-Zagier bivariate generating function enumerates the gluings by $n$ and the genus $g$ of the attendant surface and leads to a recurrence equation for the counts of gluings with parameters $n$ and $g$. This formula was originally obtained by using the multidimensional Gaussian integrals. Soon after Jackson and later Zagier found alternative proofs that used the symmetric group characters. In this note we give a different, characters-based, proof. Its core is computing and marginally inverting Fourier transform of the underlying probability measure on $S_{2n}$. Aside from Murnaghan-Nakayama rule for one-hook diagrams, the counting techniques we use are of elementary, combinatorial nature.

math.CO↗

arXiv2015

On a surface formed by randomly gluing together polygonal discs

Starting with a collection of $n$ oriented polygonal discs, with an even number $N$ of sides in total, we generate a random oriented surface by randomly matching the sides of discs and properly gluing them together. Encoding the surface in a random permutation $γ$ of $[N]$, we use the Fourier transform on $S_N$ to show that $γ$ is asymptotic to the permutation distributed uniformly on the alternating group $A_N$ ($A_N^c$ resp.) if $N-n$ and $N/2$ are of the same (opposite resp.) parity. We use this to prove a local central limit theorem for the number of vertices on the surface, whence for its Euler characteristic $χ$. We also show that with high probability the random surface consists of a single component, and thus has a well-defined genus $g=1-χ/2$, which is asymptotic to a Gaussian random variable, with mean $(N/2-n-\log N)/2$ and variance $(\log N)/2$.

math.CO↗

arXiv2014

Formation of a giant component in the intersection graph of a random chord diagram

We study the number of chords and the number of crossings in the largest component of a random chord diagram when the chords are sparsely crossing. This is equivalent to studying the number of vertices and the number of edges in the largest component of the random intersection graph. Denoting the number of chords by n and the number of crossings by m, when m/nlog(n) tends to a limit in (0,2/π^2), we show that the chord diagram chosen uniformly at random from all the diagrams with given parameters has a component containing almost all the crossings and a positive fraction of chords. On the other hand, when m < n/14, the size of the largest component is of size O(log n). One of the key analytical ingredients is an asymptotic expression for the number of chord diagrams with parameters n and m for m <(2/π^2)n\log(n), based on the Touchard-Riordan formula and the Jacobi identity for Euler partition function.

math.CO↗

arXiv2014

Distance between two random k-out digraphs, with and without preferential attachment

A random k-out mapping (digraph) on [n] is generated by choosing k random images of each vertex one at a time, subject to a "preferential attachment" rule: the current vertex selects an image i with probability proportional to a given parameter α= α(n) plus the number of times i has already been selected. Intuitively, the larger αgets, the closer the resulting k-out mapping is to the uniformly random k-out mapping. We prove that α= Θ(n^{1/2}) is the threshold for αgrowing "fast enough" to make the random digraph approach the uniformly random digraph in terms of the total variation distance. We also determine an exact limit for this distance for α= βn^{1/2}.

math.CO↗

arXiv2014

The Satisfiability Threshold for k-XORSAT

We consider "unconstrained" random $k$-XORSAT, which is a uniformly random system of $m$ linear non-homogeneous equations in $\mathbb{F}_2$ over $n$ variables, each equation containing $k \geq 3$ variables, and also consider a "constrained" model where every variable appears in at least two equations. Dubois and Mandler proved that $m/n=1$ is a sharp threshold for satisfiability of constrained 3-XORSAT, and analyzed the 2-core of a random 3-uniform hypergraph to extend this result to find the threshold for unconstrained 3-XORSAT. We show that $m/n=1$ remains a sharp threshold for satisfiability of constrained $k$-XORSAT for every $k\ge 3$, and we use standard results on the 2-core of a random $k$-uniform hypergraph to extend this result to find the threshold for unconstrained $k$-XORSAT. For constrained $k$-XORSAT we narrow the phase transition window, showing that $m-n \to -\infty$ implies almost-sure satisfiability, while $m-n \to +\infty$ implies almost-sure unsatisfiability.

math.CO↗

arXiv2013

The Satisfiability Threshold for $k$-XORSAT, using an alternative proof

We consider "unconstrained" random $k$-XORSAT, which is a uniformly random system of $m$ linear non-homogeneous equations in $\mathbb{F}_2$ over $n$ variables, each equation containing $k \ge 3$ variables, and also consider a "constrained" model where every variable appears in at least two equations. Dubois and Mandler proved that $m/n=1$ is a sharp threshold for satisfiability of constrained 3-XORSAT, and analyzed the 2-core of a random 3-uniform hypergraph to extend this result to find the threshold for unconstrained 3-XORSAT. We show that $m/n=1$ remains a sharp threshold for satisfiability of constrained $k$-XORSAT for every $k \ge 3$, and we use standard results on the 2-core of a random $k$-uniform hypergraph to extend this result to find the threshold for unconstrained $k$-XORSAT. For constrained $k$-XORSAT we narrow the phase transition window, showing that $n-m \to \infty$ implies almost-sure satisfiability, while $m-n \to \infty$ implies almost-sure unsatisfiability.

math.CO↗

arXiv2013

On the connected components of a random permutation graph with a given number of edges

A permutation of [n] induces a graph on [n] such that the edges of the graph correspond to inversion pairs of the permutation. This graph is connected if and only if the corresponding permutation is indecomposable. Let s(n,m) denote a permutation chosen uniformly at random among all permutations of [n] with exactly m inversions. Let p(n,m) be the common value for the probabilities that s(n,m) is indecomposable or the corresponding graph is connected. We prove that p(n,m) is non-decreasing with m by constructing a Markov process in which s(n,m+1) is obtained from s(n,m) by increasing one of the components of the inversion sequence of s(n,m) by one. We show that, with probability approaching 1, the graph corresponding to s(n,m) becomes connected for m asymptotic to (6/(π^2))nln(n). More precisely, for m=(6n/(π^2)) [ln(n)+ lnln(n)/2+ ln(12)- ln(π)- 12/(π^2)+x_n], where |x_n|=o(lnlnln(n)), the number of components of the random graph is shown to be asymptotically 1+Poisson(e^{-x_n}). When x_n goes to negative infinity, the sizes of the largest and the smallest components, scaled by n, are asymptotic to the lengths of the largest and the smallest subintervals in a partition of [0,1] by [e^{-x_n}] randomly, and independently, scattered points.

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