arXiv subjects

Joel Spencer

Publications and source records attributed to Joel Spencer.

At least 19 recordsLinked to original sources

arXiv2018

Preferential Attachment When Stable

We study an urn process with two urns, initialized with a ball each. Balls are added sequentially, the urn being chosen independently with probability proportional to the $α^{th}$ power $(α>1)$ of the existing number of balls. We study the (rare) event that the urn compositions are balanced after the addition of $2n-2$ new balls. We derive precise asymptotics of the probability of this event by embedding the process in continuous time. Quite surprisingly, a fine control on this probability may be leveraged to derive a lower tail Large Deviation Principle (LDP) for $L = \sum_{i=1}^{n} \frac{S_i^2}{i^2}$, where $\{S_n : n \geq 0\}$ is a simple symmetric random walk started at zero. We provide an alternate proof of the LDP via coupling to Brownian motion, and subsequent derivation of the LDP for a continuous time analogue of $L$. Finally, we turn our attention back to the urn process conditioned to be balanced, and provide a functional limit law describing the trajectory of the urn process.

math.PR↗

arXiv2017

Second order logic on random rooted trees

We address questions of logic and expressibility in the context of random rooted trees. Infiniteness of a rooted tree is not expressible as a first order sentence, but is expressible as an existential monadic second order sentence (EMSO). On the other hand, finiteness is not expressible as an EMSO. For a broad class of random tree models, including Galton-Watson trees with offspring distributions that have full support, we prove the stronger statement that finiteness does not agree up to a null set with any EMSO. We construct a finite tree and a non-null set of infinite trees that cannot be distinguished from each other by any EMSO of given parameters. This is proved via set-pebble Ehrenfeucht games (where an initial colouring round is followed by a given number of pebble rounds).

math.PR↗

arXiv2016

First Order Probabilities For Galton-Watson Trees

In the regime of Galton-Watson trees, first order logic statements are roughly equivalent to examining the presence of specific finite subtrees. We consider the space of all trees with Poisson offspring distribution and show that such finite subtrees will be almost surely present when the tree is infinite. Introducing the notion of universal trees, we show that all first order sentences of quantifier depth $k$ depend only on local neighbourhoods of the root of sufficiently large radius depending on $k$. We compute the probabilities of these neighbourhoods conditioned on the tree being infinite. We give an almost sure theory for infinite trees.

math.PR↗

arXiv2016

Galton-Watson Probability Contraction

We are concerned with exploring the probabilities of first order statements for Galton-Watson trees with $Poisson(c)$ offspring distribution. Fixing a positive integer $k$, we exploit the $k$-move Ehrenfeucht game on rooted trees for this purpose. Let $Σ$, indexed by $1 \leq j \leq m$, denote the finite set of equivalence classes arising out of this game, and $D$ the set of all probability distributions over $Σ$. Let $x_{j}(c)$ denote the true probability of the class $j \in Σ$ under $Poisson(c)$ regime, and $\vec{x}(c)$ the true probability vector over all the equivalence classes. Then we are able to define a natural recursion function $Γ$, and a map $Ψ= Ψ_{c}: D \rightarrow D$ such that $\vec{x}(c)$ is a fixed point of $Ψ_{c}$, and starting with any distribution $\vec{x} \in D$, we converge to this fixed point via $Ψ$ because it is a contraction. We show this both for $c \leq 1$ and $c > 1$, though the techniques for these two ranges are quite different.

math.PR↗

arXiv2015

Heat diffusion with frozen boundary

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.

math.PR↗

arXiv2014

Queuing with future information

We study an admissions control problem, where a queue with service rate $1-p$ receives incoming jobs at rate $λ\in(1-p,1)$, and the decision maker is allowed to redirect away jobs up to a rate of $p$, with the objective of minimizing the time-average queue length. We show that the amount of information about the future has a significant impact on system performance, in the heavy-traffic regime. When the future is unknown, the optimal average queue length diverges at rate $\sim\log_{1/(1-p)}\frac{1}{1-λ}$, as $λ\to 1$. In sharp contrast, when all future arrival and service times are revealed beforehand, the optimal average queue length converges to a finite constant, $(1-p)/p$, as $λ\to1$. We further show that the finite limit of $(1-p)/p$ can be achieved using only a finite lookahead window starting from the current time frame, whose length scales as $\mathcal{O}(\log\frac{1}{1-λ})$, as $λ\to1$. This leads to the conjecture of an interesting duality between queuing delay and the amount of information about the future.

math.PR↗

arXiv2013

On the length of a random minimum spanning tree

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.

math.CO↗

arXiv2012

The Bohman-Frieze Process Near Criticality

The Erdős-Rényi process begins with an empty graph on n vertices and edges are added randomly one at a time to a graph. A classical result of Erdős and Rényi states that the Erdős-Rényi process undergoes a phase transition, which takes place when the number of edges reaches n/2 (we say at time 1) and a giant component emerges. Since this seminal work of Erdős and Rényi, various random graph models have been introduced and studied. In this paper we study the so-called Bohman-Frieze process, a simple modification of the Erdős-Rényi process. The Bohman-Frieze process begins with an empty graph on n vertices. At each step two random edges are present and if the first edge would join two isolated vertices, it is added to a graph; otherwise the second edge is added. We present several new results on the phase transition of the Bohman-Frieze random graph process. We show that the Bohman-Frieze process has a qualitatively similar phase transition to the Erdős-Rényi process in terms of the size and structure of the components near the critical point. We prove that all components at time t_c-\eps (that is, when the number of edges are (t_c-\eps) n/2) are trees or unicyclic components and that the largest component is of size Ω(\eps^{-2} \log n). Further, at t_c + \eps, all components apart from the giant component are trees or unicyclic and the size of the second-largest component is Θ(\eps^{-2} \log n). Each of these results corresponds to an analogous well-known result for the Erdős-Rényi process. Our methods include combinatorial arguments and a combination of the differential equation method for random processes with singularity analysis of generating functions which satisfy quasi-linear partial differential equations.

math.CO↗

arXiv2010

Deterministic Random Walks on Regular Trees

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 $\exp(Ω(D^2))$ vertices contribute by being occupied by a number of chips not divisible by $k$ at a certain time.

math.CO↗

arXiv2010

Phase transitions for modified Erdös-Rényi processes

A fundamental and very well studied region of the Erdös-Rényi process is the phase transition at 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.

math.CO↗

arXiv2009

The second largest component in the supercritical 2D Hamming graph

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 $2(n-1)p=1+ε$. Previous work by van der Hofstad and Luczak had shown that in the barely supercritical region $n^{-2/3}\ln^{1/3}n\ll ε\ll 1$ the largest component has size $\sim 2εn$. Here we show that the second largest component has size close to $ε^{-2}$, 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.

math.PR↗

arXiv2006

Deterministic Random Walks on the Integers

Jim Propp's P-machine, also known as the "rotor router model" is a simple deterministic process that simulates a random walk on a graph. Instead of distributing chips to randomly chosen neighbors, it serves the neighbors in a fixed order. We investigate how well this process simulates a random walk. For the graph being the infinite path, we show that, independent of the starting configuration, at each time and on each vertex, the number of chips on this vertex deviates from the expected number of chips in the random walk model by at most a constant c_1, which is approximately 2.29. For intervals of length L, this improves to a difference of O(log L), for the L_2 average of a contiguous set of intervals even to O(sqrt{log L}). All these bounds are tight.

math.CO↗

arXiv2005

Definitions with no quantifier alternation

Let $D(G)$ be the minimum quantifier depth of a first order sentence $Φ$ 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 $Φ$. 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.

math.LO↗

arXiv2005

Avoiding defeat in a balls-in-bins process with feedback

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 α<1/2 of the balls; in fact, this probability is \exp(c_p(α)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.

math.PR↗

arXiv2005

Connectivity transitions in networks with super-linear preferential attachment

We analyze an evolving network model of Krapivsky and Redner in which new nodes arrive sequentially, each connecting to a previously existing node b with probability proportional to the p-th power of the in-degree of b. We restrict to the super-linear case p>1. When 1+1/k< p \leq 1 + 1/(k-1) the structure of the final countable tree is determined. There is a finite tree T with distinguished v (which has a limiting distribution) on which is "glued" a specific infinite tree. v has an infinite number of children, an infinite number of which have k-1 children, and there are only a finite number of nodes (possibly only v) with k or more children. Our basic technique is to embed the discrete process in a continuous time process using exponential random variables, a technique that has previously been employed in the study of balls-in-bins processes with feedback.

math.PR↗

arXiv2005

First Order Definability of Trees and Sparse Random Graphs

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 first order descriptive complexity of G. We will show that almost surely D(G)=Θ(\ln n/\ln\ln n), 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 l, so we study this problem as well.

math.CO↗

arXiv2005

A point process describing the component sizes in the critical window of the random graph evolution

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.

math.PR↗

arXiv2005

Counting Connected Graphs Asymptotically

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.

math.CO↗