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Neal Madras

Publications and source records attributed to Neal Madras.

At least 19 recordsLinked to original sources
arXiv2026

A central limit theorem for the signatures of 2-bridge knots

Cohen, Lowrance, Madras, and Raanes computed the average (absolute value of) signature over all 2-bridge knots with crossing number $c$ by introducing the number $s(c,\sigma)$ of 2-bridge knots of crossing number $c$ and signature $\sigma$. Here we provide a closed formula for this number. We use these calculations to show that the distribution of the signatures of 2-bridge knots with crossing number $c$ approaches a normal distribution as $c$ tends to infinity.

math.GT↗
arXiv2024

Average signature and 4-genus of 2-bridge knots

We show that the average or expected absolute value of the signatures of all 2-bridge knots with crossing number $c$ approaches $\sqrt{{2c}/{\pi}}$. Baader, Kjuchukova, Lewark, Misev, and Ray consider a model for 2-bridge knot diagrams indexed by diagrammatic crossing number $n$ and show that the average 4-genus is sublinear in $n$. We build upon this result in two ways to obtain an upper bound for the average 4-genus of a 2-bridge knot: our model is indexed by crossing number $c$ and gives a specific sublinear upper bound of $9.75c/\log c$.

math.GT↗
arXiv2024

A Nordhaus--Gaddum problem for the spectral gap of a graph

Let $G$ be a graph on $n$ vertices, with complement $\overline{G}$. The spectral gap of the transition probability matrix of a random walk on $G$ is used to estimate how fast the random walk becomes stationary. We prove that the larger spectral gap of $G$ and $\overline{G}$ is $\Omega(1/n)$. Moreover, if all degrees are $\Omega(n)$ and $n-\Omega(n)$, then the larger spectral gap of $G$ and $\overline{G}$ is $\Theta(1)$. We also show that if the maximum degree is $n-O(1)$ or if $G$ is a join of two graphs, then the spectral gap of $G$ is $\Omega(1/n)$. Finally, we provide a family of connected graphs with connected complements such that the larger spectral gap of $G$ and $\overline{G}$ is $O(1/n^{3/4})$.

math.CO↗
arXiv2023

Bounds on Kemeny's constant of a graph and the Nordhaus-Gaddum problem

We study Nordhaus-Gaddum problems for Kemeny's constant $\mathcal{K}(G)$ of a connected graph $G$. We prove bounds on $\min\{\mathcal{K}(G),\mathcal{K}(\overline{G})\}$ and the product $\mathcal{K}(G)\mathcal{K}(\overline{G})$ for various families of graphs. In particular, we show that if the maximum degree of a graph $G$ on $n$ vertices is $n-O(1)$ or $n-\Omega(n)$, then $\min\{\mathcal{K}(G),\mathcal{K}(\overline{G})\}$ is at most $O(n)$.

math.CO↗
arXiv2022

Adsorption of Lattice Polymers with Quenched Topologies

We introduce a framework for adsorption of a single polymer in which the topology of the polymer is quenched before adsorption, in contrast to more standard adsorption models having annealed topology. Our "topology" refers either to the precise branching structure of a branched polymer (in any dimension), or else to the knot type of a ring polymer in three dimensions. The quenched topology is chosen uniformly at random from all lattice polymers of a given size in one of four classes (lattice animals, trees, combs, or rings), and we then consider adsorption of the subclass of configurations that have the quenched topology. When the polymer-surface attraction increases without bound, the quenched topological structure keeps a macroscopic fraction of monomers off the surface, in contrast with annealed models that asymptotically have 100% of monomers in the surface. We prove properties of the limiting free energy and the critical point in each model, although important open questions remain. We pay special attention to the class of comb polymers, which admit some rigorous answers to questions that otherwise remain open. Since the class of all combs was not previously examined rigorously in full generality, we also prove the existence of its growth constant and its limiting free energy for annealed adsorption.

math-ph↗
arXiv2020

Bounded affine permutations II. Avoidance of decreasing patterns

We continue our study of a new boundedness condition for affine permutations, motivated by the fruitful concept of periodic boundary conditions in statistical physics. We focus on bounded affine permutations of size $N$ that avoid the monotone decreasing pattern of fixed size $m$. We prove that the number of such permutations is asymptotically equal to $(m-1)^{2N} N^{(m-2)/2}$ times an explicit constant as $N\to\infty$. For instance, the number of bounded affine permutations of size $N$ that avoid $321$ is asymptotically equal to $4^N (N/4\pi)^{1/2}$. We also prove a permuton-like result for the scaling limit of random permutations from this class, showing that the plot of a typical bounded affine permutation avoiding $m\cdots1$ looks like $m-1$ random lines of slope $1$ whose $y$ intercepts sum to $0$.

math.CO↗
arXiv2020

epidemic dynamics and adaptive vaccination strategy: renewal equation approach

We use analytical methods to investigate a continuous vaccination strategy effects on the infectious disease dynamics in a closed population and a demographically open population. The methodology and key assumptions are based on Breda et al (2012). We show that the cumulative force of infection for the closed population and the endemic force of infection in the demographically open population can be reduced significantly by combining two factors: the vaccine effectiveness and the vaccination rate. The impact of these factors on the force of infection can transform an endemic steady state into a disease-free state. Keywords: Force of infection, Cumulative force of infection, Scalar-renewal equation, Per capita death rate, Lambert function, adaptive vaccination strategy

q-bio.PE↗
arXiv2020

Bounded affine permutations I. Pattern avoidance and enumeration

We introduce a new boundedness condition for affine permutations, motivated by the fruitful concept of periodic boundary conditions in statistical physics. We study pattern avoidance in bounded affine permutations. In particular, we show that if $\tau$ is one of the finite increasing oscillations, then every $\tau$-avoiding affine permutation satisfies the boundedness condition. We also explore the enumeration of pattern-avoiding affine permutations that can be decomposed into blocks, using analytic methods to relate their exact and asymptotic enumeration to that of the underlying ordinary permutations. Finally, we perform exact and asymptotic enumeration of the set of all bounded affine permutations of size $n$. A companion paper will focus on avoidance of monotone decreasing patterns in bounded affine permutations.

math.CO↗
arXiv2017

Directed polymers on a disordered tree with a defect subtree

We study the question of how the competition between $\textit{bulk disorder}$ and a $\textit{localized microscopic defect}$ affects the macroscopic behavior of a system in the directed polymer context at the free energy level. We consider the directed polymer model on a disordered $d$-ary tree and represent the localized microscopic defect by modifying the disorder distribution at each vertex in a single path (branch), or in a subtree, of the tree. The polymer must choose between following the microscopic defect and finding the best branches through the bulk disorder. We describe three possible phases, called the $\textit{fully pinned, partially pinned}$ and $\textit{depinned}$ phases. When the microscopic defect is associated only with a single branch, we compute the free energy and the critical curve of the model, and show that the partially pinned phase does not occur. When the localized microscopic defect is associated with a non-disordered regular subtree of the disordered tree, the picture is more complicated. We prove that all three phases are non-empty below a critical temperature, and that the partially pinned phase disappears above the critical temperature.

math.PR↗
arXiv2016

Longest monotone subsequences and rare regions of pattern-avoiding permutations

We consider the distributions of the lengths of the longest monotone and alternating subsequences in classes of permutations of size $n$ that avoid a specific pattern or set of patterns, with respect to the uniform distribution on each such class. We obtain exact results for any class that avoids two patterns of length 3, as well as results for some classes that avoid one pattern of length 4 or more. In our results, the longest monotone subsequences have expected length proportional to $n$ for pattern-avoiding classes, in contrast with the $\sqrt n$ behaviour that holds for unrestricted permutations. In addition, for a pattern $\tau$ of length $k$, we scale the plot of a random $\tau$-avoiding permutation down to the unit square and study the "rare region," which is the part of the square that is exponentially unlikely to contain any points. We prove that when $\tau_1>\tau_k$, the complement of the rare region is a closed set that contains the main diagonal of the unit square. For the case $\tau_1=k,$ we also show that the lower boundary of the part of the rare region above the main diagonal is a curve that is Lipschitz continuous and strictly increasing on $[0,1]$.

math.CO↗
arXiv2016

Location of the Adsorption Transition for Lattice Polymers

We consider various lattice models of polymers: lattice trees, lattice animals, and self-avoiding walks. The polymer interacts with a surface (hyperplane), receiving a unit energy reward for each site in the surface. There is an adsorption transition of the polymer at a critical value of $\beta$, the inverse temperature. We present a new proof of the result of Hammersley, Torrie, and Whittington (1982) that the transition occurs at a strictly positive value of $\beta$ when the surface is impenetrable, i.e. when the polymer is restricted to a half-space. In contrast, for a penetrable surface, it is an open problem to prove that the transition occurs at $\beta=0$ (i.e., infinite temperature). We reduce this problem to showing that the fraction of N-site polymers whose span is less than $N/\log^2 N$ is not too small.

math-ph↗
arXiv2016

Large Deviations for Permutations Avoiding Monotone Patterns

For a given permutation $\tau$, let $P_N^{\tau}$ be the uniform probability distribution on the set of $N$-element permutations $\sigma$ that avoid the pattern $\tau$. For $\tau=\mu_k:=123\cdots k$, we consider $P_N^{\mu_k}(\sigma_I=J)$ where $I\sim \gamma N$ and $J\sim \delta N$ for $\gamma,\delta \in (0,1)$. If $\gamma+\delta\neq 1$ then we are in the large deviations regime with the probability decaying exponentially, and we calculate the limiting value of $P_N^{\mu_k}(\sigma_I=J)^{1/N}$. We also observe that for $\tau = \lambda_{k,\ell} := 12\ldots\ell k(k-1)\ldots(\ell+1)$ and $\gamma+\delta<1$, the limit of $P_N^{\tau}(\sigma_I=J)^{1/N}$ is the same as for $\tau=\mu_k$.

math.CO↗
arXiv2015

A Note on Diffusion State Distance

Diffusion state distance (DSD) is a metric on the vertices of a graph, motivated by bioinformatic modeling. Previous results on the convergence of DSD to a limiting metric relied on the definition being based on symmetric or reversible random walk on the graph. We show that convergence holds even when the DSD is based on general finite irreducible Markov chains. The proofs rely on classical potential theory of Kemeny and Snell.

math.PR↗
arXiv2014

Stability of adversarial Markov chains, with an application to adaptive MCMC algorithms

We consider whether ergodic Markov chains with bounded step size remain bounded in probability when their transitions are modified by an adversary on a bounded subset. We provide counterexamples to show that the answer is no in general, and prove theorems to show that the answer is yes under various additional assumptions. We then use our results to prove convergence of various adaptive Markov chain Monte Carlo algorithms.

math.PR↗
arXiv2014

Convergence Rates for Hierarchical Gibbs Samplers

We establish some results for the rate of convergence in total variation of a Gibbs sampler to its equilibrium distribution. This sampler is motivated by a hierarchical Bayesian inference construction for a gamma random variable. Our results apply to a wide range of parameter values in the case that the hierarchical depth is 3 or 4, and are more restrictive for depth greater than 4. Our method involves showing a relationship between the total variation of two ordered copies of our chain and the maximum of the ratios of their respective co-ordinates. We construct auxiliary stochastic processes to show that this ratio does converge to 1 at a geometric rate.

math.PR↗
arXiv2014

Structure of Random 312-Avoiding Permutations

We evaluate the probabilities of various events under the uniform distribution on the set of 312-avoiding permutations of 1,...,N. We derive exact formulas for the probability that the ith element of a random permutation is a specific value less than i, and for joint probabilities of two such events. In addition, we obtain asymptotic approximations to these probabilities for large N when the elements are not close to the boundaries or to each other. We also evaluate the probability that the graph of a random 312-avoiding permutation has k specified decreasing points, and we show that for large N the points below the diagonal look like trajectories of a random walk.

math.PR↗
arXiv2011

Quantitative bounds for Markov chain convergence: Wasserstein and total variation distances

We present a framework for obtaining explicit bounds on the rate of convergence to equilibrium of a Markov chain on a general state space, with respect to both total variation and Wasserstein distances. For Wasserstein bounds, our main tool is Steinsaltz's convergence theorem for locally contractive random dynamical systems. We describe practical methods for finding Steinsaltz's "drift functions" that prove local contractivity. We then use the idea of "one-shot coupling" to derive criteria that give bounds for total variation distances in terms of Wasserstein distances. Our methods are applied to two examples: a two-component Gibbs sampler for the Normal distribution and a random logistic dynamical system.

math.ST↗