arXiv subjects

Serban Nacu

Publications and source records attributed to Serban Nacu.

6 recordsLinked to original sources

arXiv2016

How round are the complementary components of planar Brownian motion?

Consider a Brownian motion $W$ in ${\bf C}$ started from $0$ and run for time 1. Let $A(1),A(2),\dots$ denote the bounded connected components of ${\bf C}-W([0,1])$. Let $R(i)$ (resp. $r(i)$) denote the out-radius (resp. in-radius) of $A(i)$ for $i\in\bf N$. Our main result is that ${\bf E}[\sum_i R(i)^2|\log R(i)|^θ]<\infty$ for any $θ<1$. We also prove that $\sum_i r(i)^2|\log r(i)|=\infty$ almost surely. These results have the interpretation that most of the components $A(i)$ have a rather regular or round shape.

math.PR↗

arXiv2010

Random soups, carpets and fractal dimensions

We study some properties of a class of random connected planar fractal sets induced by a Poissonian scale-invariant and translation-invariant point process. Using the second-moment method, we show that their Hausdorff dimensions are deterministic and equal to their expectation dimension. We also estimate their low-intensity limiting behavior. This applies in particular to the "conformal loop ensembles" defined via Poissonian clouds of Brownian loops for which the expectation dimension has been computed by Schramm, Sheffield and Wilson.

math.PR↗

arXiv2005

Channel Uncertainty in Ultra Wideband Communication Systems

Wide band systems operating over multipath channels may spread their power over bandwidth if they use duty cycle. Channel uncertainty limits the achievable data rates of power constrained wide band systems; Duty cycle transmission reduces the channel uncertainty because the receiver has to estimate the channel only when transmission takes place. The optimal choice of the fraction of time used for transmission depends on the spectral efficiency of the signal modulation. The general principle is demonstrated by comparing the channel conditions that allow different modulations to achieve the capacity in the limit. Direct sequence spread spectrum and pulse position modulation systems with duty cycle achieve the channel capacity, if the increase of the number of channel paths with the bandwidth is not too rapid. The higher spectral efficiency of the spread spectrum modulation lets it achieve the channel capacity in the limit, in environments where pulse position modulation with non-vanishing symbol time cannot be used because of the large number of channel paths.

cs.IT↗

arXiv2005

Fast simulation of new coins from old

Let S\subset (0,1). Given a known function f:S\to (0,1), we consider the problem of using independent tosses of a coin with probability of heads p (where p\in S is unknown) to simulate a coin with probability of heads f(p). We prove that if S is a closed interval and f is real analytic on S, then f has a fast simulation on S (the number of p-coin tosses needed has exponential tails). Conversely, if a function f has a fast simulation on an open set, then it is real analytic on that set.

math.PR↗

arXiv2004

Increments of Random Partitions

For any partition of $\{1, 2, ..., n\}$ we define its {\it increments} $X_i, 1 \le i \le n$ by $X_i = 1$ if $i$ is the smallest element in the partition block that contains it, $X_i = 0$ otherwise. We prove that for partially exchangeable random partitions (where the probability of a partition depends only on its block sizes in order of appearance), the law of the increments uniquely determines the law of the partition. One consequence is that the Chinese Restaurant Process CRP($θ$) (the partition with distribution given by the Ewens sampling formula with parameter $θ$) is the only exchangeable random partition with independent increments.

math.PR↗

arXiv2003

Glauber Dynamics On The Cycle Is Monotone

We study heat-bath Glauber dynamics for the ferromagnetic Ising model on a finite cycle (a graph where every vertex has degree two). We prove that the relaxation time $τ_2$ is an increasing function of any of the couplings $J_{xy}$. We also prove some further inequalities, and obtain exact asymptotics for $τ_2$ at low temperatures.

math.PR↗