[Show abstract][Hide abstract] 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; DOI:10.1007/s10955-015-1342-6 · 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
[Show abstract][Hide abstract] 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.
[Show abstract][Hide abstract] 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
[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):1-25. DOI:10.1007/s11512-011-0157-1 · 0.95 Impact Factor
[Show abstract][Hide abstract] 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; 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 $\log^*n-\log^*\log^*n-1\le q_0(n)\le \log^*n+22$, 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(8):2264-2283. 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 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.
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 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.
[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 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.
[Show abstract][Hide abstract] 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.
[Show abstract][Hide abstract] 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; 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 $k-1+l$ 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; 27(8). DOI:10.1016/j.ejc.2006.05.006 · 0.65 Impact Factor
[Show abstract][Hide abstract] 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.
[Show abstract][Hide abstract] 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
[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
$\log^*n-\log^*\log^*n-1\le q_0(n)\le \log^*n+22$, 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.
[Show abstract][Hide abstract] 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; 15(06). DOI:10.1017/S0963548306007565 · 0.62 Impact Factor