arXiv subjects

Martin Hildebrand

Publications and source records attributed to Martin Hildebrand.

7 recordsLinked to original sources

arXiv2021

A multiplicatively symmetrized version of the Chung-Diaconis-Graham random process

This paper considers random processes of the form $X_{n+1}=a_nX_n+b_n \pmod p$ where $p$ is odd, $X_0=0$, $(a_0,b_0), (a_1,b_1), (a_2,b_2),...$ are i.i.d., and $a_n$ and $b_n$ are independent with $P(a_n=2)=P(a_n=(p+1)/2)=1/2$ and $P(b_n=1)=P(b_n=0)=P(b_n=-1)=1/3$. This can be viewed as a multiplicatively symmetrized version of a random process of Chung, Diaconis, and Graham. This paper shows that order $(\log p)^2$ steps suffice for $X_n$ to be close to uniformly distributed on the integers mod $p$ for all odd $p$ while order $(\log p)^2$ steps are necessary for $X_n$ to be close to uniformly distributed on the integers mod $p$.

math.PR↗

arXiv2015

A Condition for Distinguishing Sceneries on Non-abelian Groups

A scenery $f$ on a finite group $G$ is a function from $G$ to $\{0,1\}$. A random walk $v(t)$ on $G$ is said to be reconstructive if the distributions of 2 sceneries evaluated on the random walk with uniform initial distribution are identical only if one scenery is a shift of the other scenery. Previous results gave a sufficient condition for reconstructivity on finite abelian groups. This paper gives a ready generalization of this sufficient condition to one for reconstructivity on finite non-abelian groups but shows that no random walks on finite non-abelian groups satisfy this sufficient condition.

math.PR↗

arXiv2008

A lower bound for the Chung-Diaconis-Graham random process

Chung, Diaconis, and Graham considered random processes of the form X_{n+1}=a_n X_n+b_n (mod p) where p is odd, X_0=0, a_n=2 always, and b_n are i.i.d. for n=0,1,2,... . In this paper, we show that if P(b_n=-1)=P{b_n=0)=P(b_n=1)=1/3, then there exists a constant c>1 such that c log_2 p steps are not enough to make X_n get close to uniformly distributed on the integers mod p.

math.PR↗

arXiv2007

Generating Random Vectors in (Z/pZ)^d Via an Affine Random Process

This paper considers some random processes of the form X_{n+1}=TX_n+B_n (mod p) where B_n and X_n are random variables over (Z/pZ)^d and T is a fixed d x d integer matrix which is invertible over the complex numbers. For a particular distribution for B_n, this paper improves results of Asci to show that if T has no complex eigenvalues of length 1, then for integers p relatively prime to det(T), order (log p)^2 steps suffice to make X_n close to uniformly distributed where X_0 is the zero vector. This paper also shows that if T has a complex eigenvalue which is a root of unity, then order p^b steps are needed for X_n to get close to uniform where b is a value which may depend on T and X_0 is the zero vector.

math.PR↗

arXiv2007

On the Chung-Diaconis-Graham random process

Chung, Diaconis, and Graham considered random processes of the form X_{n+1}=2X_n+b_n (mod p) where X_0=0, p is odd, and b_n for n=0,1,2,... are i.i.d. random variables on {-1,0,1}. If Pr(b_n=-1)= Pr(b_n=1)=βand Pr(b_n=0)=1-2β, they asked which value of βmakes X_n get close to uniformly distributed on the integers mod p the slowest. In this paper, we extend the results of Chung, Diaconis, and Graham in the case p=2^t-1 to show that for 0<β<=1/2, there is no such value of β.

math.PR↗

arXiv2004

Bounding quantities related to the packing density of 1(L+1)L...2

We bound several quantities related to the packing density of the patterns 1(L+1)L...2. These bounds sharpen results of Bóna, Sagan, and Vatter and give a new proof of the packing density of these patterns, originally computed by Stromquist in the case L=2 and by Price for larger L. We end with comments and conjectures.

math.CO↗

arXiv2004

A Survey of Results on Random Random Walks on Finite Groups

A number of papers have examined various aspects of "random random" walks on finite groups; the purpose of this article is to provide a survey of this work and to show, bring together, and discuss some of the arguments and results in this work. This article also provides a number of exercises. Some exercises involve straightforward computations; others involve proving details in proofs or extending results proved in the article. This article also describes some problems for further study.

math.PR↗