arXiv subjects
Publications and source records attributed to David A. Levin.
We present a Markov chain on the $n$-dimensional hypercube $\{0,1\}^n$ which satisfies $t_{\rm mix}(ε) = n[1 + o(1)]$. This Markov chain alternates between random and deterministic moves and we prove that the chain has cut-off with a window of size at most $O(n^{0.5+δ})$ where $δ>0$. The deterministic moves correspond to a linear shift register.
We define and study a model of winding for non-colliding particles in finite trees. We prove that the asymptotic behavior of this statistic satisfies a central limiting theorem, analogous to similar results on winding of bounded particles in the plane. We also propose certain natural open questions and conjectures, whose confirmation would provide new insights on configuration spaces of trees.
The spectral gap $γ$ of a finite, ergodic, and reversible Markov chain is an important parameter measuring the asymptotic rate of convergence. In applications, the transition matrix $P$ may be unknown, yet one sample of the chain up to a fixed time $n$ may be observed. We consider here the problem of estimating $γ$ from this data. Let $π$ be the stationary distribution of $P$, and $π_\star = \min_x π(x)$. We show that if $n = \tilde{O}\bigl(\frac{1}{γπ_\star}\bigr)$, then $γ$ can be estimated to within multiplicative constants with high probability. When $π$ is uniform on $d$ states, this matches (up to logarithmic correction) a lower bound of $\tildeΩ\bigl(\frac{d}γ\bigr)$ steps required for precise estimation of $γ$. Moreover, we provide the first procedure for computing a fully data-dependent interval, from a single finite-length trajectory of the chain, that traps the mixing time $t_{\text{mix}}$ of the chain at a prescribed confidence level. The interval does not require the knowledge of any parameters of the chain. This stands in contrast to previous approaches, which either only provide point estimates, or require a reset mechanism, or additional prior knowledge. The interval is constructed around the relaxation time $t_{\text{relax}} = 1/γ$, which is strongly related to the mixing time, and the width of the interval converges to zero roughly at a $1/\sqrt{n}$ rate, where $n$ is the length of the sample path.
The spectral gap $γ$ of an ergodic and reversible Markov chain is an important parameter measuring the asymptotic rate of convergence. In applications, the transition matrix $P$ may be unknown, yet one sample of the chain up to a fixed time $t$ may be observed. Hsu, Kontorovich, and Szepesvari (2015) considered the problem of estimating $γ$ from this data. Let $π$ be the stationary distribution of $P$, and $π_\star = \min_x π(x)$. They showed that, if $t = \tilde{O}\bigl(\frac{1}{γ^3 π_\star}\bigr)$, then $γ$ can be estimated to within multiplicative constants with high probability. They also proved that $\tildeΩ\bigl(\frac{n}γ\bigr)$ steps are required for precise estimation of $γ$. We show that $\tilde{O}\bigl(\frac{1}{γπ_\star}\bigr)$ steps of the chain suffice to estimate $γ$ up to multiplicative constants with high probability. When $π$ is uniform, this matches (up to logarithmic corrections) the lower bound of Hsu, Kontorovich, and Szepesvari.
We analyze the mixing behavior of the biased exclusion process on a path of length $n$ as the bias $β_n$ tends to $0$ as $n \to \infty$. We show that the sequence of chains has a pre-cutoff, and interpolates between the unbiased exclusion and the process with constant bias. As the bias increases, the mixing time undergoes two phase transitions: one when $β_n$ is of order $1/n$, and the other when $β_n$ is order $\log n/n$.
In this paper, we study a family of lattice walks which are related to the Hadamard conjecture. There is a bijection between paths of these walks which originate and terminate at the origin and equivalence classes of partial Hadamard matrices. Therefore, the existence of partial Hadamard matrices can be proved by showing that there is positive probability of a random walk returning to the origin after a specified number of steps. Moreover, the number of these designs can be approximated by estimating the return probabilities. We use the inversion formula for the Fourier transform of the random walk to provide such estimates. We also include here an upper bound, derived by elementary methods, on the number of partial Hadamard.
Let X^{(k)}(t) = (X_1(t), ..., X_k(t)) denote a k-vector of i.i.d. random variables, each taking the values 1 or 0 with respective probabilities p and 1-p. As a process indexed by non-negative t, $X^{(k)}(t)$ is constructed--following Benjamini, Haggstrom, Peres, and Steif (2003)--so that it is strong Markov with invariant measure ((1-p)δ_0+pδ_1)^k. We derive sharp estimates for the probability that ``X_1(t)+...+X_k(t)=k-\ell for some t in F,'' where F \subset [0,1] is nonrandom and compact. We do this in two very different settings: (i) Where \ell is a constant; and (ii) Where \ell=k/2, k is even, and p=q=1/2. We prove that the probability is described by the Kolmogorov capacitance of F for case (i) and Howroyd's 1/2-dimensional box-dimension profiles for case (ii). We also present sample-path consequences, and a connection to capacities that answers a question of Benjamini et. al. (2003)
We study the Glauber dynamics for the Ising model on the complete graph, also known as the Curie-Weiss Model. For beta < 1, we prove that the dynamics exhibits a cut-off: the distance to stationarity drops from near 1 to near 0 in a window of order n centered at [2(1-beta)]^{-1} n log n. For beta = 1, we prove that the mixing time is of order n^{3/2}. For beta > 1, we study metastability. In particular, we show that the Glauber dynamics restricted to states of non-negative magnetization has mixing time O(n log n).
We derive a new coupling of the running maximum of an Ornstein-Uhlenbeck process and the running maximum of an explicit i.i.d. sequence. We use this coupling to verify a conjecture of Darling and Erdos (1956).
We present an extreme-value analysis of the classical law of the iterated logarithm (LIL) for Brownian motion. Our result can be viewed as a new improvement to the LIL.
We propose a set-indexed family of capacities $\{\cap_G \}_{G \subseteq \R_+}$ on the classical Wiener space $C(\R_+)$. This family interpolates between the Wiener measure ($\cap_{\{0\}}$) on $C(\R_+)$ and the standard capacity ($\cap_{\R_+}$) on Wiener space. We then apply our capacities to characterize all quasi-sure lower functions in $C(\R_+)$. In order to do this we derive the following capacity estimate which may be of independent interest: There exists a constant $a > 1$ such that for all $r > 0$, \[ \frac {1}{a} \K_G(r^6) e^{-π^2/(8r^2)} \le \cap_G \{f^* \le r\} \le a \K_G(r^6) e^{-π^2/(8r^2)}. \] Here, $\K_G$ denotes the Kolmogorov $ε$-entropy of $G$, and $f^* := \sup_{[0,1]}|f|$.
Consider a sequence {X(i,0) : i = 1, ..., n} of i.i.d. random variables. Associate to each X(i,0) an independent mean-one Poisson clock. Every time a clock rings replace that X-variable by an independent copy. In this way, we obtain i.i.d. stationary processes {X(i,t) : t >= 0} (i=1,2, ...) whose invariant distribution is the law of X(1,0). Benjamini, Haggstrom, Peres, and Steif (2003) introduced the dynamical walk S(n,t) = X(1,t) + ... + X(n,t), and proved among other things that the LIL holds for {S(n,t) : n =1,2, ...} simultaneously for all t. In other words, the LIL is dynamically stable. Subsequently, we showed that in the case that the X(i,0)'s are standard normal, the classical integral test is not dynamically stable. Presently, we study the set of times t when {S(n,t) : n=1,2, ...} exceeds a given envelope infinitely often. Our analysis is made possible thanks to a connection to the Kolmogorov epsilon-entropy. When used in conjunction with the invariance principle of this paper, this connection has other interesting by-products some of which we relate. We prove also that viewed as an infinite-dimensional process, the rescaled dynamical random walk converges weakly in D(D([0,1])) to the Ornstein-Uhlenbeck process in C([0,1]). For this we assume only that the increments have mean zero and variance one. In addition, we extend a result of Benjamini, Haggstrom, Peres and Steif (2003) by proving that if the X(i,0)'s are lattice, mean-zero variance-one, and possess 2 + epsilon finite absolute moments for some positive epsilon, then the recurrence of the origin is dynamically stable. To prove this we derive a gambler's ruin estimate that is valid for all lattice random walks that have mean zero and finite variance. We believe the latter may be of independent interest.
Suppose that a coin with bias theta is tossed at renewal times of a renewal process, and a fair coin is tossed at all other times. Let mu_θbe the distribution of the observed sequence of coin tosses, and let u_n denote the chance of a renewal at time n. Harris and Keane showed that if sum_{n=1}^infty u_n^2=\infty, then mu_theta and μ_0 are singular, while if sum_{n=1}^{infty} u_n^2 theta_c, they are singular. We also prove that when u_n=O(n^{-1}), the measures mu_theta for theta in [-1,1] are all mutually absolutely continuous.
Suppose that attached to each site z in Z is a coin with bias theta(z), and only finitely many of these coins have non-zero bias. Allow a simple random walker to generate observations by tossing, at each move, the coin attached to its current position. Then we can determine the biases {theta(z) : z in Z}, using only the outcomes of these coin tosses and no information about the path of the random walker, up to a shift and reflection of Z. This generalizes a result of Harris and Keane.