Publications (49)25.14 Total impact

Article: Heat Diffusion with Frozen Boundary
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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.Journal of Statistical Physics 03/2015; 161(3). DOI:10.1007/s1095501513426 · 1.20 Impact Factor  [Show abstract] [Hide abstract]
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.Combinatorics Probability and Computing 08/2012; DOI:10.1017/S0963548315000024 · 0.62 Impact Factor 
Article: Proppian random walks in Z
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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(5):349361. DOI:10.1016/j.disc.2010.11.001 · 0.56 Impact Factor 
Conference Paper: Deterministic Discrepancy Minimization.
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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 nonconstructive entropy method is used to argue the feasibility of this enhanced SDP.Algorithms  ESA 2011  19th Annual European Symposium, Saarbrücken, Germany, September 59, 2011. Proceedings; 01/2011  [Show abstract] [Hide abstract]
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 kary 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., 2010Random Structures and Algorithms 10/2010; 37(3). DOI:10.1002/rsa.20314 · 0.92 Impact Factor  [Show abstract] [Hide abstract]
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):125. DOI:10.1007/s1151201101571 · 0.95 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The 2dimensional 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 $2(n1)p=1+\epsilon$. 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; 36(1). DOI:10.1002/rsa.v36:1 · 0.92 Impact Factor  [Show abstract] [Hide abstract]
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 $\log^*n\log^*\log^*n1\le q_0(n)\le \log^*n+22$, 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(8):22642283. DOI:10.1016/j.ejc.2007.04.016 · 0.65 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Let D(G) be the smallest quantifier depth of a firstorder formula which is true for a graph G but false for any other nonisomorphic graph. This can be viewed as a measure for the descriptive complexity of G in firstorder 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.Combinatorics Probability and Computing 04/2007; 16(03):375  400. DOI:10.1017/S0963548306008376 · 0.62 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We study random subgraphs of the ncube {0,1}n, where nearestneighbor 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 ncube.Combinatorica 07/2006; 26(4):395410. DOI:10.1007/s0049300600221 · 0.70 Impact Factor 
Conference Paper: Deterministic random walks
Proceedings of the Eighth Workshop on Algorithm Engineering and Experiments and the Third Workshop on Analytic Algorithmics and Combinatorics; 01/2006  [Show abstract] [Hide abstract]
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 pth 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 wellapproximated by the solution to a certain ordinary differential equation. 
Article: Discrepancy Games.
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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 nuniform 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 09/2005; 12(1). · 0.49 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We introduce and analyze a liar game in which tary 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):537559. DOI:10.1007/s0049300500333 · 0.70 Impact Factor 
Article: A Point Process Describing the Component Sizes in the Critical Window of the Random Graph Evolution
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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 = n1 + λn4/3, 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.Combinatorics Probability and Computing 06/2005; 16(04). DOI:10.1017/S0963548306008327 · 0.62 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We find the asymptotic number of connected graphs with $k$ vertices and $k1+l$ edges when $k,l$ approach infinity, reproving a result of Bender, Canfield and McKay. We use the {\em probabilistic method}, analyzing breadthfirst 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; 27(8). DOI:10.1016/j.ejc.2006.05.006 · 0.65 Impact Factor 
Article: Birth Control for Giants
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ABSTRACT: The standard ErdosRényi 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.Combinatorica 07/2004; 27(5). DOI:10.1007/s0049300721632 · 0.70 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: nstep simple random walk from its starting point. At time n, we expect that the Pmachine 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 Pmachine 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 DMS0303272. #( # n) chips at position v = 0. The deterministic Pmachine will give that number with only constant error. As a further interpretation, consider a "linear machine" in  [Show abstract] [Hide abstract]
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 $\log^*n\log^*\log^*n1\le q_0(n)\le \log^*n+22$, 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.  [Show abstract] [Hide abstract]
ABSTRACT: We analyze Jim Propp's Pmachine, 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 signchanges 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; 15(06). DOI:10.1017/S0963548306007565 · 0.62 Impact Factor
Publication Stats
601  Citations  
25.14  Total Impact Points  
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Institutions

19942015

CUNY Graduate Center
New York, New York, United States


2011

NYU Langone Medical Center
New York, New York, United States


19952005

Mathematical Sciences Research Institute
Berkeley, California, United States


2000

Alfréd Rényi Institute of Mathematics
Budapeŝto, Budapest, Hungary
