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Ohad N. Feldheim

Publications and source records attributed to Ohad N. Feldheim.

14 recordsLinked to original sources
arXiv2022

The umpteen operator and its Lifshitz tails

As put forth by Kerov in the early 1990s and elucidated in subsequent works, numerous properties of Wigner random matrices are shared by certain linear maps playing an important rôle in the representation theory of the symmetric group. We introduce and study an operator of representation-theoretic origin which bears some similarity to discrete random Schrödinger operators acting on the $d$-dimensional lattice. In particular, we define its integrated density of states and prove that in dimension $d \geq 2$ it boasts Lifshitz tails similar to those of the Anderson model. The construction is closely related to an infinite-board version of the fifteen puzzle, a popular sliding puzzle from the XIX-th century. We estimate, using a new Peierls argument, the probability that the puzzle returns to its initial state after $n$ random moves. The Lifshitz tail is deduced using an identification of our random operator with the action of the adjacency matrix of the puzzle on a randomly chosen representation of the infinite symmetric group.

math-ph↗
arXiv2021

Min-cost-flow preserving bijection between subgraphs and orientations

Consider an undirected graph $G=(V,E)$. A subgraph of $G$ is a subset of its edges, whilst an orientation of $G$ is an assignment of a direction to each edge. Provided with an integer circulation-demand $d:V\to \mathbb{Z}$, we show an explicit and efficiently computable bijection between subgraphs of $G$ on which a $d$-flow exists and orientations on which a $d$-flow exists. Moreover, given a cost function $w:E\to (0,\infty)$ we can find such a bijection which preserves the $w$-min-cost-flow. In 2013, Kozma and Moran showed, using dimensional methods, that the number of subgraphs $k$-connecting a vertex $s$ to a vertex $t$ is the same as the number of orientations $k$-connecting $s$ to $t$. An application of our result is an efficient, bijective proof of this fact.

math.CO↗
arXiv2021

Long-term balanced allocation via thinning

We study the long-term behavior of the two-thinning variant of the classical balls-and-bins model. In this model, an overseer is provided with uniform random allocation of $m$ balls into $n$ bins in an on-line fashion. For each ball, the overseer could reject its allocation and place the ball into a new bin drawn independently at random. The purpose of the overseer is to reduce the maximum load of the bins, which is defined as the difference between the maximum number of balls in a single bin and $m/n$, i.e., the average number of balls among all bins. We provide tight estimates for three quantities: the lowest maximum load that could be achieved at time $m$, the lowest maximum load that could be achieved uniformly over the entire time interval $[m]:=\{1, 2, \cdots, m\}$, and the lowest \emph{typical} maximum load that could be achieved over the interval $[m]$, where the typicality means that the maximum load holds for $1-o(1)$ portion of the times in $[m]$. We show that when $m$ and $n$ are sufficiently large, a typical maximum load of $(\log n)^{1/2+o(1)}$ can be achieved with high probability, asymptotically the same as the optimal maximum load that could be achieved at time $m$. However, for any strategy, the maximal load among all times in the interval $[m]$ is $Ω\big(\frac{\log n}{\log\log n}\big)$ with high probability. A strategy achieving this bound is provided. An explanation for this gap is provided by our optimal strategies as follows. To control the typical load, we restrain the maximum load for some time, during which we accumulate more and more bins with relatively high load. After a while, we have to employ for a short time a different strategy to reduce the number of relatively heavily loaded bins, at the expanse of temporarily inducing high load in a few bins.

math.PR↗
arXiv2021

Drawing outerplanar graphs using thirteen edge lengths

We show that every outerplanar graph $G$ can be linearly embedded in the plane such that the number of distinct distances between pairs of adjacent vertices is at most thirteen and there is no intersection between the image of a vertex and that of an edge not containing it. This extends the work of Alon and the second author, where only overlap between vertices was disallowed, thus settling a problem posed by Carmi, Dujmović, Morin and Wood.

math.CO↗
arXiv2020

Load balancing under $d$-thinning

In the classical balls-and-bins model, $m$ balls are allocated into $n$ bins one by one uniformly at random. In this note, we consider the $d$-thinning variant of this model, in which the process is regulated in an on-line fashion as follows. For each ball, after a random bin has been selected, an overseer may decide, based on all previous history, whether to accept this bin or not. However, one of every $d$ consecutive suggested bins must be accepted. The maximum load of this setting is the number of balls in the most loaded bin. We show that after $Θ(n)$ balls have been allocated, the least maximum load achievable with high probability is $(d+o(1))\sqrt[d]{\frac{d\log n}{\log\log n}}$. This should be compared with the related $d$-choice setting, in which the optimal maximum load achievable with high probability is $\frac{\log\log n}{\log d}+O(1)$.

math.PR↗
arXiv2018

The power of thinning in balanced allocation

Balls are sequentially allocated into $n$ bins as follows: for each ball, an independent, uniformly random bin is generated. An overseer may then choose to either allocate the ball to this bin, or else the ball is allocated to a new independent uniformly random bin. The goal of the overseer is to reduce the load of the most heavily loaded bin after $Θ(n)$ balls have been allocated. We provide an asymptotically optimal strategy yielding a maximum load of $(1+o(1))\sqrt{\frac{8\log n}{\log\log n}}$ balls.

math.PR↗
arXiv2017

The power of online thinning in reducing discrepancy

Consider an infinite sequence of independent, uniformly chosen points from $[0,1]^d$. After looking at each point in the sequence, an overseer is allowed to either keep it or reject it, and this choice may depend on the locations of all previously kept points. However, the overseer must keep at least one of every two consecutive points. We call a sequence generated in this fashion a \emph{two-thinning} sequence. Here, the purpose of the overseer is to control the discrepancy of the empirical distribution of points, that is, after selecting $n$ points, to reduce the maximal deviation of the number of points inside any axis-parallel hyper-rectangle of volume $A$ from $nA$. Our main result is an explicit low complexity two-thinning strategy which guarantees discrepancy of $O(\log^{2d+1} n)$ for all $n$ with high probability (compare with $Θ(\sqrt{n\log\log n})$ without thinning). The case $d=1$ of this result answers a question of Benjamini. We also extend the construction to achieve the same asymptotic bound for ($1+β$)-thinning, a set-up in which rejecting is only allowed with probability $β$ independently for each point. In addition, we suggest an improved and simplified strategy which we conjecture to guarantee discrepancy of $O(\log^{d+1} n)$ (compare with $θ(\log^d n)$, the best known construction of a low discrepancy sequence). Finally, we provide theoretical and empirical evidence for our conjecture, and provide simulations supporting the viability of our construction for applications.

math.PR↗
arXiv2017

Rigidity of 3-colorings of the discrete torus

We prove that a uniformly chosen proper $3$-coloring of the $d$-dimensional discrete torus has a very rigid structure when the dimension $d$ is sufficiently high. We show that with high probability the coloring takes just one color on almost all of either the even or the odd sub-torus. In particular, one color appears on nearly half of the torus sites. This model is the zero temperature case of the $3$-state anti-ferromagnetic Potts model from statistical physics. Our work extends previously obtained results for the discrete torus with specific boundary conditions. The main challenge in this extension is to overcome certain topological obstructions which appear when no boundary conditions are imposed on the model. Locally, a proper $3$-coloring defines the discrete gradient of an integer-valued height function which changes by exactly one between adjacent sites. However, these locally-defined functions do not always yield a height function on the entire torus, as the gradients may accumulate to a non-zero quantity when winding around the torus. Our main result is that in high dimensions, a global height function is well defined with high probability, allowing to deduce the rigid structure of the coloring from previously known results. Moreover, the probability that the gradients accumulate to a vector $m$, corresponding to the winding in each of the $d$ directions, is at most exponentially small in the product of $\|m\|_\infty$ and the area of a cross-section of the torus. In the course of the proof we develop discrete analogues of notions from algebraic topology. This theory is developed in some generality and may be of use in the study of other models.

math-ph↗
arXiv2016

Double roots of random polynomials with integer coefficients

We consider random polynomials whose coefficients are independent and identically distributed on the integers. We prove that if the coefficient distribution has bounded support and its probability to take any particular value is at most $\tfrac12$, then the probability of the polynomial to have a double root is dominated by the probability that either $0$, $1$, or $-1$ is a double root up to an error of $o(n^{-2})$. We also show that if the support of coefficient distribution excludes $0$ then the double root probability is $O(n^{-2})$. Our result generalizes a similar result of Peled, Sen and Zeitouni for Littlewood polynomials.

math.PR↗
arXiv2014

A note on general sliding window processes

Let $f:\mathbb{R}^k\to \mathbb{R}$ be a measurable function, and let $\{U_i\}_{i\in\mathbb{N}}$ be a sequence of i.i.d. random variables. Consider the random process $Z_i=f(U_{i},...,U_{i+k-1})$. We show that for all $\ell$, there is a positive probability, uniform in $f$, for $Z_1,...,Z_\ell$ to be monotone. We give upper and lower bounds for this probability, and draw corollaries for $k$-block factor processes with a finite range. The proof is based on an application of combinatorial results from Ramsey theory to the realm of continuous probability.

math.PR↗
arXiv2013

Long gaps between sign-changes of Gaussian Stationary Processes

We study the probability of a real-valued stationary process to be positive on a large interval $[0,N]$. We show that if in some neighborhood of the origin the spectral measure of the process has density which is bounded away from zero and infinity, then the decay of this probability is bounded between two exponential functions in $N$. This generalizes similar bounds obtained for particular cases, such as a recent result by Artezana, Buckley, Marzo, Olsen.

math.PR↗
arXiv2010

3/2 Firefighters are not enough

The firefighter problem is a monotone dynamic process in graphs that can be viewed as modeling the use of a limited supply of vaccinations to stop the spread of an epidemic. In more detail, a fire spreads through a graph, from burning vertices to their unprotected neighbors. In every round, a small amount of unburnt vertices can be protected by firefighters. How many firefighters per turn, on average, are needed to stop the fire from advancing? We prove tight lower and upper bounds on the amount of firefighters needed to control a fire in the Cartesian planar grid and in the strong planar grid, resolving two conjectures of Ng and Raff.

cs.DM↗
arXiv2009

One more proof of the Erdős--Turán inequality, and an error estimate in Wigner's law

We prove the Erdos-Turan equidistribution inequality, using a construction due to Chebyshev, Markov, and Stieltjes. The method is applicable in a more general setting. As an example, we state another inequality that can be proved using this method. The inequality can be used to get an error estimate in Wigner's law which is sharp near the edge.

math.CA↗