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Tobias Johnson

Publications and source records attributed to Tobias Johnson.

29 records · Page 2Linked to original sources
arXiv2017

Local limit of the fixed point forest

Consider the following partial "sorting algorithm" on permutations: take the first entry of the permutation in one-line notation and insert it into the position of its own value. Continue until the first entry is 1. This process imposes a forest structure on the set of all permutations of size $n$, where the roots are the permutations starting with 1 and the leaves are derangements. Viewing the process in the opposite direction towards the leaves, one picks a fixed point and moves it to the beginning. Despite its simplicity, this "fixed point forest" exhibits a rich structure. In this paper, we consider the fixed point forest in the limit $n\to \infty$ and show using Stein's method that at a random permutation the local structure weakly converges to a tree defined in terms of independent Poisson point processes. We also show that the distribution of the length of the longest path to a leaf converges to the geometric distribution with mean $e-1$, and the length of the shortest path converges to the Poisson distribution with mean 1. In addition, the higher moments are bounded and hence the expectations converge as well.

math.PR↗
arXiv2016

From transience to recurrence with Poisson tree frogs

Consider the following interacting particle system on the $d$-ary tree, known as the frog model: Initially, one particle is awake at the root and i.i.d. Poisson many particles are sleeping at every other vertex. Particles that are awake perform simple random walks, awakening any sleeping particles they encounter. We prove that there is a phase transition between transience and recurrence as the initial density of particles increases, and we give the order of the transition up to a logarithmic factor.

math.PR↗
arXiv2016

Recurrence and transience for the frog model on trees

The frog model is a growing system of random walks where a particle is added whenever a new site is visited. A longstanding open question is how often the root is visited on the infinite $d$-ary tree. We prove the model undergoes a phase transition, finding it recurrent for $d=2$ and transient for $d\geq 5$. Simulations suggest strong recurrence for $d=2$, weak recurrence for $d=3$, and transience for $d\geq 4$. Additionally, we prove a 0-1 law for all $d$-ary trees, and we exhibit a graph on which a 0-1 law does not hold. To prove recurrence when $d=2$, we construct a recursive distributional equation for the number of visits to the root in a smaller process and show the unique solution must be infinity a.s. The proof of transience when $d=5$ relies on computer calculations for the transition probabilities of a large Markov chain. We also include the proof for $d \geq 6$, which uses similar techniques but does not require computer assistance.

math.PR↗
arXiv2015

Quantitative Small Subgraph Conditioning

We revisit the method of small subgraph conditioning, used to establish that random regular graphs are Hamiltonian a.a.s. We refine this method using new technical machinery for random $d$-regular graphs on $n$ vertices that hold not just asymptotically, but for any values of $d$ and $n$. This lets us estimate how quickly the probability of containing a Hamiltonian cycle converges to 1, and it produces quantitative contiguity results between different models of random regular graphs. These results hold with $d$ held fixed or growing to infinity with $n$. As additional applications, we establish the distributional convergence of the number of Hamiltonian cycles when $d$ grows slowly to infinity, and we prove that the number of Hamiltonian cycles can be approximately computed from the graph's eigenvalues for almost all regular graphs.

math.PR↗
arXiv2015

Exchangeable pairs, switchings, and random regular graphs

We consider the distribution of cycle counts in a random regular graph, which is closely linked to the graph's spectral properties. We broaden the asymptotic regime in which the cycle counts are known to be approximately Poisson, and we give an explicit bound in total variation distance for the approximation. Using this result, we calculate limiting distributions of linear eigenvalue functionals for random regular graphs. Previous results on the distribution of cycle counts by McKay, Wormald, and Wysocka (2004) used the method of switchings, a combinatorial technique for asymptotic enumeration. Our proof uses Stein's method of exchangeable pairs and demonstrates an interesting connection between the two techniques.

math.CO↗
arXiv2014

The Marčenko-Pastur law for sparse random bipartite biregular graphs

We prove that the empirical spectral distribution of a (d_L, d_R)-biregular, bipartite random graph, under certain conditions, converges to a symmetrization of the Marčenko-Pastur distribution of random matrix theory. This convergence is not only global (on fixed-length intervals) but also local (on intervals of increasingly smaller length). Our method parallels the one used previously by Dumitriu and Pal (2012).

math.PR↗
arXiv2014

Cycles and eigenvalues of sequentially growing random regular graphs

Consider the sum of $d$ many i.i.d. random permutation matrices on $n$ labels along with their transposes. The resulting matrix is the adjacency matrix of a random regular (multi)-graph of degree $2d$ on $n$ vertices. It is known that the distribution of smooth linear eigenvalue statistics of this matrix is given asymptotically by sums of Poisson random variables. This is in contrast with Gaussian fluctuation of similar quantities in the case of Wigner matrices. It is also known that for Wigner matrices the joint fluctuation of linear eigenvalue statistics across minors of growing sizes can be expressed in terms of the Gaussian Free Field (GFF). In this article, we explore joint asymptotic (in $n$) fluctuation for a coupling of all random regular graphs of various degrees obtained by growing each component permutation according to the Chinese Restaurant Process. Our primary result is that the corresponding eigenvalue statistics can be expressed in terms of a family of independent Yule processes with immigration. These processes track the evolution of short cycles in the graph. If we now take $d$ to infinity, certain GFF-like properties emerge.

math.PR↗
arXiv2014

Eigenvalue fluctuations for random regular graphs

One of the major themes of random matrix theory is that many asymptotic properties of traditionally studied distributions of random matrices are universal. We probe the edges of universality by studying the spectral properties of random regular graphs. Specifically, we prove limit theorems for the fluctuations of linear spectral statistics of random regular graphs. We find both universal and non-universal behavior. Our most important tool is Stein's method for Poisson approximation, which we develop for use on random regular graphs. This is my Ph.D. thesis, based on joint work with Ioana Dumitriu, Elliot Paquette, and Soumik Pal. For the most part, it's a mashed up version of arXiv:1109.4094, arXiv:1112.0704, and arXiv:1203.1113, but some things in here are improved or new. In particular, Chapter 4 goes into more detail on some of the proofs than arXiv:1203.1113 and includes a new section. See Section 1.3 for more discussion on what's new and who contributed to what.

math.PR↗
arXiv2012

Functional limit theorems for random regular graphs

Consider d uniformly random permutation matrices on n labels. Consider the sum of these matrices along with their transposes. The total can be interpreted as the adjacency matrix of a random regular graph of degree 2d on n vertices. We consider limit theorems for various combinatorial and analytical properties of this graph (or the matrix) as n grows to infinity, either when d is kept fixed or grows slowly with n. In a suitable weak convergence framework, we prove that the (finite but growing in length) sequences of the number of short cycles and of cyclically non-backtracking walks converge to distributional limits. We estimate the total variation distance from the limit using Stein's method. As an application of these results we derive limits of linear functionals of the eigenvalues of the adjacency matrix. A key step in this latter derivation is an extension of the Kahn-Szemerédi argument for estimating the second largest eigenvalue for all values of d and n.

math.PR↗
arXiv2008

On Universal Cycles for Multisets

A Universal Cycle for t-multisets of [n]={1,...,n} is a cyclic sequence of $\binom{n+t-1}{t}$ integers from [n] with the property that each t-multiset of [n] appears exactly once consecutively in the sequence. For such a sequence to exist it is necessary that n divides $\binom{n+t-1}{t}$, and it is reasonable to conjecture that this condition is sufficient for large enough n in terms of t. We prove the conjecture completely for t in {2,3} and partially for t in {4,6}. These results also support a positive answer to a question of Knuth.

math.CO↗