Joel Spencer

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

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Publications (46)23.25 Total impact

  • 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.
    Discrete Mathematics 03/2011; 311:349-361. · 0.57 Impact Factor
  • 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.
    Algorithms - ESA 2011 - 19th Annual European Symposium, Saarbrücken, Germany, September 5-9, 2011. Proceedings; 01/2011
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    ABSTRACT: Jim Propp's rotor router model is a deterministic analogue 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. (2006)) show a remarkable similarity of both models. If an (almost) arbitrary population of chips is placed on the vertices of a grid $\Z^d$ 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 ($k \ge 3$), 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 $\exp(\Omega(D^2))$ vertices contribute by being occupied by a number of chips not divisible by $k$ at a certain time. Comment: 15 pages, to appear in Random Structures and Algorithms
    Random Structures and Algorithms 06/2010; · 1.05 Impact Factor
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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.
    Arkiv för matematik 05/2010; 50(2):1-25. · 0.60 Impact Factor
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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.
    Random Structures and Algorithms 02/2008; · 1.05 Impact Factor
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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.
    European Journal of Combinatorics 11/2007; 28:2264-2283. · 0.61 Impact Factor
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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 , where G is a random tree of order n or the giant component of a random graph with constant c<1. These results rely on computing the maximum of D(T) for a tree T of order n and maximum degree l, so we study this problem as well.
    Combinatorics Probability and Computing 04/2007; 16(03):375 - 400. · 0.61 Impact Factor
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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.
    Combinatorica 07/2006; 26(4):395-410. · 0.63 Impact Factor
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    Proceedings of the Eighth Workshop on Algorithm Engineering and Experiments and the Third Workshop on Analytic Algorithmics and Combinatorics; 01/2006
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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.
    11/2005;
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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.
    Combinatorica 08/2005; 25(5):537-559. · 0.63 Impact Factor
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    Svante Janson, Joel Spencer
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    ABSTRACT: We study a point process describing the asymptotic behavior of sizes of the largest components of the random graph G(n,p) in the critical window p=n^{-1}+lambda n^{-4/3}. 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 epsilon is almost constant.
    Combinatorics Probability and Computing 06/2005; · 0.61 Impact Factor
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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.
    European Journal of Combinatorics 03/2005; · 0.61 Impact Factor
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    ABSTRACT: From the text: In the classical theory of Maker/Breaker positional games a hypergraph H=(V,E) is given and the players, Maker and Breaker, take turns in occupying a previously unoccupied element of the “board” V. The goal of Breaker is to prevent Maker from fully occupying an edge. The well-known criterion of P. Erdős and J. L. Selfridge [J. Combinat. Theory, Ser. A 14, 298–301 (1973; Zbl 0293.05004)] provides a strategy for Breaker to win. In a more general setting, the criterion of J. Beck [Combinatorica 1, 103–116 (1981; Zbl 0519.05005)] ensures that Breaker can select more than α|e| elements of each edge e∈E, for some α≥0. In these Maker/Breaker-type games Breaker does not care about fully occupying an edge himself. In the so-called Avoider/Forcer-type games this is the only thing player “Avoider” cares about not doing. More precisely Avoider wins the game against Forcer if at the end of the game he does not occupy any edge. More generally, Avoider wins if he occupies less than (1-α)|e| elements from any edge e∈E, for some α≥0. X. Lu [Discrete Math. 94, No. 3, 199–207 (1991; Zbl 0758.05085)] obtained criteria similar to the ones of Erdős and Selfridge, and of Beck for this case. In the present paper we investigate a game where one of the players, called Balancer, must achieve the goals of both Breaker and Avoider. Balancer’s main difficulty is that while as Breaker he cannot hurt himself by selecting any particular vertex, as Avoider he can. Similarly, as Avoider he cannot hurt himself by not occupying any particular vertex while as Breaker he can. The classical Erdős-Selfridge-type criteria do not immediately generalize to this setting. In this paper 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.
    The electronic journal of combinatorics 01/2005; 12. · 0.57 Impact Factor
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    Joel Spencer, Nicholas Wormald
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    ABSTRACT: The standard Erdos-Renyi model of random graphs begins with n isolated vertices, and at each round a random edge is added. Parametrizing rounds as one time unit, a phase transition occurs at t = 1 when a giant component (one of size constant time n) first appears. Under the influence of statistical mechanics, the investigation of related phase transitions has become an important topic in random graph theory.
    Combinatorica 07/2004; · 0.63 Impact Factor
  • 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
    05/2004;
  • 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.
    05/2004;
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    Joshua N. Cooper, Joel Spencer
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    ABSTRACT: We analyze Jim Propp's P-machine, a simple deterministic process that simulates a random walk on $Z^d$ to within a constant. The proof of the error bound relies on several estimates in the theory of simple random walks and some careful summing. We mention three intriguing conjectures concerning sign-changes and unimodality of functions in the linear span of $\{p(\cdot,x) : x \in Z^d\}$, where $p(n,x)$ is the probability that a walk beginning from the origin arrives at $x$ at time $n$.
    Combinatorics Probability and Computing 03/2004; · 0.61 Impact Factor
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    Oleg Pikhurko, Joel Spencer, Oleg Verbitsky
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    ABSTRACT: We say that a first order sentence A defines a graph G if A is true on G but false on any graph non-isomorphic to G. Let L(G) (resp. D(G)) denote the minimum length (resp. quantifier rank) of a such sentence. We define the succinctness function s(n) (resp. its variant q(n)) to be the minimum L(G) (resp. D(G)) over all graphs on n vertices. We prove that s(n) and q(n) may be so small that for no general recursive function f we can have f(s(n))\ge n for all n. However, for the function q^*(n)=\max_{i\le n}q(i), which is the least monotone nondecreasing function bounding q(n) from above, we have q^*(n)=(1+o(1))\log^*n, where \log^*n equals the minimum number of iterations of the binary logarithm sufficient to lower n below 1. We show an upper bound q(n)<\log^*n+5 even under the restriction of the class of graphs to trees. Under this restriction, for q(n) we also have a matching lower bound. We show a relationship D(G)\ge(1-o(1))\log^*L(G) and prove, using the upper bound for q(n), that this relationship is tight. For a non-negative integer a, let D_a(G) and q_a(n) denote the analogs of D(G) and q(n) for defining formulas in the negation normal form with at most a quantifier alternations in any sequence of nested quantifiers. We show a superrecursive gap between D_0(G) and D_3(G) and hence between D_0(G) and D(G). Despite it, for q_0(n) we still have a kind of log-star upper bound: q_0(n)\le2\log^*n+O(1) for infinitely many n.
    Annals of Pure and Applied Logic 02/2004; · 0.45 Impact Factor
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    ABSTRACT: In a previous paper, we defined a version of the percolation triangle condition that is suitable for the analysis of bond percolation on a finite connected transitive graph, and showed that this triangle condition implies that the percolation phase transition has many features in common with the phase transition on the complete graph. In this paper, we use a new and simplified approach to the lace expansion to prove quite generally that for finite graphs that are tori the triangle condition for percolation is implied by a certain triangle condition for simple random walks on the graph. The latter is readily verified for several graphs with vertex set $\{0,1,..., r-1\}^n$, including the Hamming cube on an alphabet of $r$ letters (the $n$-cube, for $r=2$), the $n$-dimensional torus with nearest-neighbor bonds and $n$ sufficiently large, and the $n$-dimensional torus with $n>6$ and sufficiently spread-out (long range) bonds. The conclusions of our previous paper thus apply to the percolation phase transition for each of the above examples.
    The Annals of Probability 02/2004; · 1.43 Impact Factor