Joel Spencer

CUNY Graduate Center, New York, New York, United States

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Publications (52)27.46 Total impact

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    ABSTRACT: Consider "Frozen Random Walk" on $\mathbb{Z}$: $n$ particles start at the origin. At any discrete time, the leftmost and rightmost $\lfloor{\frac{n}{4}}\rfloor$ particles are "frozen" and do not move. The rest of the particles in the "bulk" independently jump to the left and right uniformly. The goal of this note is to understand the limit of this process under scaling of mass and time. To this end we study the following deterministic mass splitting process: start with mass $1$ at the origin. At each step the extreme quarter mass on each side is "frozen". The remaining "free" mass in the center evolves according to the discrete heat equation. We establish diffusive behavior of this mass evolution and identify the scaling limit under the assumption of its existence. It is natural to expect the limit to be a truncated Gaussian. A naive guess for the truncation point might be the $1/4$ quantile points on either side of the origin. We show that this is not the case and it is in fact determined by the evolution of the second moment of the mass distribution.
    Preview · Article · Mar 2015 · Journal of Statistical Physics
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    Colin Cooper · Alan Frieze · Nate Ince · Svante Janson · Joel Spencer
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    ABSTRACT: We study the expected value of the length $L_n$ of the minimum spanning tree of the complete graph $K_n$ when each edge $e$ is given an independent uniform $[0,1]$ edge weight. We sharpen the result of Frieze \cite{F1} that $\lim_{n\to\infty}\E(L_n)=\z(3)$ and show that $\E(L_n)=\z(3)+\frac{c_1}{n}+\frac{c_2+o(1)}{n^{4/3}}$ where $c_1,c_2$ are explicitly defined constants.
    Full-text · Article · Aug 2012 · Combinatorics Probability and Computing
  • Nikhil Bansal · Joel Spencer
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    ABSTRACT: We derandomize a recent algorithmic approach due to Bansal (Foundations of Computer Science, FOCS, pp. 3–10, 2010) to efficiently compute low discrepancy colorings for several problems, for which only existential results were previously known. In particular, we give an efficient deterministic algorithm for Spencer’s six standard deviations result (Spencer in Trans. Am. Math. Soc. 289:679–706, 1985), and to find a low discrepancy coloring for a set system with low hereditary discrepancy. The main new idea is to add certain extra constraints to the natural semidefinite programming formulation for discrepancy, which allow us to argue about the existence of a good deterministic move at each step of the algorithm. The non-constructive entropy method is used to argue the feasibility of this enhanced SDP.
    No preview · Conference Paper · Sep 2011
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    Juliana Freire · Joel Spencer
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    ABSTRACT: The Propp Machine is a deterministic process that simulates a random walk. Instead of distributing chips randomly, each position makes the chips move according to the walk’s possible steps in a fixed order. A random walk is called Proppian if at each time at each position the number of chips differs from the expected value by at most a constant, independent of time or the initial configuration of chips.The simple walk where the possible steps are 1 or −1−1 each with probability p=12 is Proppian, with constant approximately 2.29. The equivalent simple walks on ZdZd are also Proppian. Here, we show the same result for a larger class of walks on ZZ, allowing an arbitrary number of possible steps with some constraint on their probabilities.
    Preview · Article · Mar 2011 · Discrete Mathematics
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    ABSTRACT: Jim Propp's rotor–router model is a deterministic analog of a random walk on a graph. Instead of distributing chips randomly, each vertex serves its neighbors in a fixed order. Cooper and Spencer (Comb Probab Comput 15 (2006) 815–822) show a remarkable similarity of both models. If an (almost) arbitrary population of chips is placed on the vertices of a grid ℤd and does a simultaneous walk in the Propp model, then at all times and on each vertex, the number of chips on this vertex deviates from the expected number the random walk would have gotten there by at most a constant. This constant is independent of the starting configuration and the order in which each vertex serves its neighbors. This result raises the question if all graphs do have this property. With quite some effort, we are now able to answer this question negatively. For the graph being an infinite k-ary tree (k ≥ 3), we show that for any deviation D there is an initial configuration of chips such that after running the Propp model for a certain time there is a vertex with at least D more chips than expected in the random walk model. However, to achieve a deviation of D it is necessary that at least exp(Ω(D2)) vertices contribute by being occupied by a number of chips not divisible by k at a certain time. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010
    Full-text · Article · Oct 2010 · Random Structures and Algorithms
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    Svante Janson · Joel Spencer
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    ABSTRACT: A fundamental and very well studied region of the Erdős–Rényi process is the phase transition at m∼n/2 edges in which a giant component suddenly appears. We examine the process beginning with an initial graph. We further examine the Bohman–Frieze process in which edges between isolated vertices are more likely. While the positions of the phase transitions vary, the three processes belong, roughly speaking, to the same universality class. In particular, the growth of the giant component in the barely supercritical region is linear in all cases.
    Preview · Article · May 2010 · Arkiv för matematik
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    ABSTRACT: The 2-dimensional Hamming graph H(2,n) consists of the $n^2$ vertices $(i,j)$, $1\leq i,j\leq n$, two vertices being adjacent when they share a common coordinate. We examine random subgraphs of H(2,n) in percolation with edge probability $p$, so that the average degree $2(n-1)p=1+\epsilon$. Previous work by van der Hofstad and Luczak had shown that in the barely supercritical region $n^{-2/3}\ln^{1/3}n\ll \epsilon \ll 1$ the largest component has size $\sim 2\epsilon n$. Here we show that the second largest component has size close to $\epsilon^{-2}$, so that the dominant component has emerged. This result also suggests that a {\it discrete duality principle} might hold, whereby, after removing the largest connected component in the supercritical regime, the remaining random subgraphs behave as in the subcritical regime.
    Full-text · Article · Feb 2008 · Random Structures and Algorithms
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    Oleg Pikhurko · Joel Spencer · Oleg Verbitsky
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    ABSTRACT: Let $D(G)$ be the minimum quantifier depth of a first order sentence $\Phi$ that defines a graph $G$ up to isomorphism. Let $D_0(G)$ be the version of $D(G)$ where we do not allow quantifier alternations in $\Phi$. Define $q_0(n)$ to be the minimum of $D_0(G)$ over all graphs $G$ of order $n$. We prove that for all $n$ we have $\log^*n-\log^*\log^*n-1\le q_0(n)\le \log^*n+22$, where $\log^*n$ is equal to the minimum number of iterations of the binary logarithm needed to bring $n$ to 1 or below. The upper bound is obtained by constructing special graphs with modular decomposition of very small depth.
    Full-text · Article · Nov 2007 · European Journal of Combinatorics
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    ABSTRACT: Let D(G) be the smallest quantifier depth of a first-order formula which is true for a graph G but false for any other non-isomorphic graph. This can be viewed as a measure for the descriptive complexity of G in first-order logic. We show that almost surely D(G) = Theta(lnn/lnlnn), where G is a random tree of order n or the giant component of a random graph G(n,c/n) with constant c > 1. These results rely on computing the maximum of D(T) for a tree T of order n and maximum degree 1, so we study this problem as well.
    Full-text · Article · Apr 2007 · Combinatorics Probability and Computing
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    ABSTRACT: We study random subgraphs of the n-cube {0,1}n, where nearest-neighbor edges are occupied with probability p. Let pc(n) be the value of p for which the expected size of the component containing a fixed vertex attains the value λ2n/3, where λ is a small positive constant. Let ε=n(p−pc(n)). In two previous papers, we showed that the largest component inside a scaling window given by |ε|=Θ(2−n/3) is of size Θ(22n/3), below this scaling window it is at most 2(log 2)nε−2, and above this scaling window it is at most O(ε2n). In this paper, we prove that for $ p - p_{c} {\left( n \right)} \geqslant e^{{cn^{{1/3}} }} $ p - p_{c} {\left( n \right)} \geqslant e^{{cn^{{1/3}} }} the size of the largest component is at least Θ(ε2n), which is of the same order as the upper bound. The proof is based on a method that has come to be known as “sprinkling,” and relies heavily on the specific geometry of the n-cube.
    Full-text · Article · Jul 2006 · Combinatorica
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    Full-text · Conference Paper · Jan 2006
  • Conference Paper: Erdős Magic
    Joel Spencer
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    ABSTRACT: The Probabilistic Method ([AS]) is a lasting legacy of the late Paul Erdős. We give two examples – both problems first formulated by Erdős in the 1960s with new results in the last decade and both with substantial open questions. Further in both examples we take a Computer Science vantagepoint, creating a probabilistic algorithm to create the object (coloring, packing, respectively) and showing that with positive probability the created object has the desired properties. Given m sets each of size n (with an arbitrary intersection pattern) we want to color the underlying vertices Red and Blue so that no set is monochromatic. Erdős showed this may always be done if m< 2n − 1 (proof: color randomly!). We give an argument of Srinivasan and Radhakrishnan ([RS]) that extends this to \(m<c2^n\sqrt{n/\ln n}\). One first colors randomly and then recolors the blemishes with a clever random sequential algorithm. In a universe of size N we have a family of sets, each of size k, such that each vertex is in D sets and any two vertices have only o(D) common sets. Asymptotics are for fixed k with N,D→∞. We want an asymptotic packing, a subfamily of ~ N/k disjoint sets. Erdős and Hanani conjectured such a packing exists (in an important special case of asymptotic designs) and this conjecture was shown by Rödl. We give a simple proof of the author ([S]) that analyzes the random greedy algorithm. Paul Erdős was a unique figure, an inspirational figure to countless mathematicians, including the author. Why did his view of mathematics resonate so powerfully? What was it that drew so many of us into his circle? Why do we love to tell Erdős stories? What was the magic of the man we all knew as Uncle Paul?
    No preview · Conference Paper · Dec 2005
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    Roberto Oliveira · Joel Spencer
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    ABSTRACT: Imagine that there are two bins to which balls are added sequentially, and each incoming ball joins a bin with probability proportional to the p-th power of the number of balls already there. A general result says that if p>1/2, there almost surely is some bin that will have more balls than the other at all large enough times, a property that we call eventual leadership. In this paper, we compute the asymptotics of the probability that bin 1 eventually leads when the total initial number of balls $t$ is large and bin 1 has a fraction \alpha<1/2 of the balls; in fact, this probability is \exp(c_p(\alpha)t + O{t^{2/3}}) for some smooth, strictly negative function c_p. Moreover, we show that conditioned on this unlikely event, the fraction of balls in the first bin can be well-approximated by the solution to a certain ordinary differential equation.
    Full-text · Article · Nov 2005
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    ABSTRACT: We investigate a game played on a hypergraph H = (V, E) by two players, Balancer and Unbalancer. They select one element of the vertex set V alternately until all vertices are selected. Balancer wins if at the end of the game all edges e ∈ E are roughly equally distributed between the two players. We give a polynomial time algorithm for Balancer to win provided the allowed deviation is large enough. In particular, it follows from our result that if H is n-uniform and has m edges, then Balancer can achieve having between n/2 - √ln(2m)n/2 and n/2 + √ln(2m)n/2 of his vertices on every edge e of H. We also discuss applications in positional game theory.
    Preview · Article · Sep 2005 · The electronic journal of combinatorics
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    Ioana Dumitriu · Joel Spencer
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    ABSTRACT: We introduce and analyze a liar game in which t-ary questions are asked and the responder may lie at most k times. As an additional constraint, there is an arbitrary but prescribed list (the channel) of permissible types of lies. For any fixed t, k, and channel, we determine the exact asymptotics of the solution when the number of queries goes to infinity.
    Full-text · Article · Aug 2005 · Combinatorica
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    Svante Janson · Joel Spencer
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    ABSTRACT: We study a point process describing the asymptotic behaviour of sizes of the largest components of the random graph G(n,p) in the critical window, that is, for p = n-1 + λn-4/3, where A is a fixed real number. In particular, we show that this point process has a surprising rigidity. Fluctuations in the large values will be balanced by opposite fluctuations in the small values such that the sum of the values larger than a small ε (a scaled version of the number of vertices in components of size greater than εn2/3) is almost constant.
    Preview · Article · Jun 2005 · Combinatorics Probability and Computing
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    Remco van der Hofstad · Joel Spencer
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    ABSTRACT: We find the asymptotic number of connected graphs with $k$ vertices and $k-1+l$ edges when $k,l$ approach infinity, reproving a result of Bender, Canfield and McKay. We use the {\em probabilistic method}, analyzing breadth-first search on the random graph $G(k,p)$ for an appropriate edge probability $p$. Central is analysis of a random walk with fixed beginning and end which is tilted to the left.
    Full-text · Article · Mar 2005 · European Journal of Combinatorics
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    Joel Spencer · Nicholas Wormald
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    ABSTRACT: The standard Erdos-Rényi model of random graphs begins with n isolated vertices, and at each round a random edge is added. Parametrizing n/2 rounds as one time unit, a phase transition occurs at time t = 1 when a giant component (one of size constant times n) first appears. Under the influence of statistical mechanics, the investigation of related phase transitions has become an important topic in random graph theory. We define a broad class of graph evolutions in which at each round one chooses one of two random edges {v 1, v 2}, {v 3, v 4} to add to the graph. The selection is made by examining the sizes of the components of the four vertices. We consider the susceptibility S(t) at time t, being the expected component size of a uniformly chosen vertex. The expected change in S(t) is found which produces in the limit a differential equation for S(t). There is a critical time t c so that S(t) → ∞ as t approaches t c from below. We show that the discrete random process asymptotically follows the differential equation for all subcritical t < t c . Employing classic results of Cramér on branching processes we show that the component sizes of the graph in the subcritical regime have an exponential tail. In particular, the largest component is only logarithmic in size. In the supercritical regime t > t c we show the existence of a giant component, so that t = t c may be fairly considered a phase transition. Computer aided solutions to the possible differential equations for susceptibility allow us to establish lower and upper bounds on the extent to which we can either delay or accelerate the birth of the giant component.
    Preview · Article · Jul 2004 · Combinatorica
  • Joshua N. Cooper · Joel Spencer
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    ABSTRACT: n-step simple random walk from its starting point. At time n, we expect that the P-machine and the random process should give rise to similar distributions if they begin with the same configuration of chips. For the random walk process we may consider the expected number of chips that will be at v at time n. Our main result is that the di#erence between this expected number and the actual number at v at time n in the deterministic P-machine is bounded uniformly -- irrespective of how much time has passed, what the original chip distribution was, the starting states of the rotors, or even the choice of v! As an example, suppose n is even, d = 1, we begin with n chips at position 0, and the total time is n. The random walk model will have an expected number n n/2 2 -n = + Research supported by NSF Grant DMS-0303272. #( # n) chips at position v = 0. The deterministic P-machine will give that number with only constant error. As a further interpretation, consider a "linear machine" in
    No preview · Article · May 2004
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    Oleg Pikhurko · Joel Spencer · Oleg Verbitsky
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    ABSTRACT: Let $D(G)$ be the minimum quantifier depth of a first order sentence $\Phi$ that defines a graph $G$ up to isomorphism. Let $D_0(G)$ be the version of $D(G)$ where we do not allow quantifier alternations in $\Phi$. Define $q_0(n)$ to be the minimum of $D_0(G)$ over all graphs $G$ of order $n$. We prove that for all $n$ we have $\log^*n-\log^*\log^*n-1\le q_0(n)\le \log^*n+22$, where $\log^*n$ is equal to the minimum number of iterations of the binary logarithm needed to bring $n$ to 1 or below. The upper bound is obtained by constructing special graphs with modular decomposition of very small depth.
    Preview · Article · May 2004

Publication Stats

666 Citations
27.46 Total Impact Points

Institutions

  • 1994-2015
    • CUNY Graduate Center
      New York, New York, United States
  • 2004-2012
    • NYU Langone Medical Center
      New York, New York, United States
  • 1995-2005
    • Mathematical Sciences Research Institute
      Berkeley, California, United States
  • 2000
    • Alfréd Rényi Institute of Mathematics
      Budapeŝto, Budapest, Hungary
  • 1996
    • City University of New York - York College
      New York, New York, United States
  • 1988
    • State University of New York
      New York, New York, United States