Alistair Sinclair
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View article: On quantum to classical comparison for Davies generators
On quantum to classical comparison for Davies generators Open
Despite extensive study, our understanding of quantum Markov chains remains far less complete than that of their classical counterparts. [Temme'13] observed that the Davies Lindbladian, a well-studied model of quantum Markov dynamics, cont…
View article: Nonlinear Dynamics for the Ising Model
Nonlinear Dynamics for the Ising Model Open
We introduce and analyze a natural class of nonlinear dynamics for spin systems such as the Ising model. This class of dynamics is based on the framework of mass action kinetics, which models the evolution of systems of entities under pair…
View article: Nonlinear Dynamics for the Ising Model
Nonlinear Dynamics for the Ising Model Open
We introduce and analyze a natural class of nonlinear dynamics for spin systems such as the Ising model. This class of dynamics is based on the framework of mass action kinetics, which models the evolution of systems of entities under pair…
View article: Diversity in Evolutionary Dynamics
Diversity in Evolutionary Dynamics Open
We consider the dynamics imposed by natural selection on the populations of two competing, sexually reproducing, haploid species. In this setting, the fitness of any genome varies over time due to the changing population mix of the competi…
View article: Spatial mixing and the random‐cluster dynamics on lattices
Spatial mixing and the random‐cluster dynamics on lattices Open
An important paradigm in the understanding of mixing times of Glauber dynamics for spin systems is the correspondence between spatial mixing properties of the models and bounds on the mixing time of the dynamics. This includes, in particul…
View article: Nonlinear dynamics for the Ising model
Nonlinear dynamics for the Ising model Open
We introduce and analyze a natural class of nonlinear dynamics for spin systems such as the Ising model. This class of dynamics is based on the framework of mass action kinetics, which models the evolution of systems of entities under pair…
View article: Correlation Decay and Partition Function Zeros: Algorithms and Phase Transitions
Correlation Decay and Partition Function Zeros: Algorithms and Phase Transitions Open
We explore connections between the phenomenon of correlation decay and the location of Lee-Yang and Fisher zeros for various spin systems. In particular we show that, in many instances, proofs showing that weak spatial mixing on the Bethe …
View article: Spatial mixing and the random-cluster dynamics on lattices
Spatial mixing and the random-cluster dynamics on lattices Open
An important paradigm in the understanding of mixing times of Glauber dynamics for spin systems is the correspondence between spatial mixing properties of the models and bounds on the mixing time of the dynamics. This includes, in particul…
View article: Low-temperature Ising dynamics with random initializations
Low-temperature Ising dynamics with random initializations Open
Glauber dynamics on spin systems are well known to suffer exponential slowdowns at low temperatures due to the emergence of multiple metastable phases, separated by narrow bottlenecks that are hard for the dynamics to cross. It is a folklo…
View article: The critical mean-field Chayes–Machta dynamics
The critical mean-field Chayes–Machta dynamics Open
The random-cluster model is a unifying framework for studying random graphs, spin systems and electrical networks that plays a fundamental role in designing efficient Markov Chain Monte Carlo (MCMC) sampling algorithms for the classical fe…
View article: Efficiently list‐edge coloring multigraphs asymptotically optimally
Efficiently list‐edge coloring multigraphs asymptotically optimally Open
We give polynomial time algorithms for the seminal results of Kahn, who showed that the Goldberg–Seymour and list‐coloring conjectures for (list‐)edge coloring multigraphs hold asymptotically. Kahn's arguments are based on the probabilisti…
View article: Low-temperature Ising dynamics with random initializations
Low-temperature Ising dynamics with random initializations Open
It is well known that Glauber dynamics on spin systems typically suffer exponential slowdowns at low temperatures. This is due to the emergence of multiple metastable phases in the state space, separated by narrow bottlenecks that are hard…
View article: The Critical Mean-Field Chayes-Machta Dynamics
The Critical Mean-Field Chayes-Machta Dynamics Open
The random-cluster model is a unifying framework for studying random graphs, spin systems and electrical networks that plays a fundamental role in designing efficient Markov Chain Monte Carlo (MCMC) sampling algorithms for the classical fe…
View article: Entropy decay in the Swendsen-Wang dynamics
Entropy decay in the Swendsen-Wang dynamics Open
We study the mixing time of the Swendsen-Wang dynamics for the ferromagnetic Ising and Potts models on the integer lattice ${\mathbb Z}^d$. This dynamics is a widely used Markov chain that has largely resisted sharp analysis because it is …
View article: Entropy decay in the Swendsen-Wang dynamics on ${\mathbb Z}^d$
Entropy decay in the Swendsen-Wang dynamics on ${\mathbb Z}^d$ Open
We study the mixing time of the Swendsen-Wang dynamics for the ferromagnetic Ising and Potts models on the integer lattice ${\mathbb Z}^d$. This dynamics is a widely used Markov chain that has largely resisted sharp analysis because it is …
View article: A Deterministic Algorithm for Counting Colorings with 2-Delta Colors
A Deterministic Algorithm for Counting Colorings with 2-Delta Colors Open
We give a polynomial time deterministic approximation algorithm (an FPTAS) for counting the number of q-colorings of a graph of maximum degree Delta, provided only that q ≥ 2Delta. This substantially improves on previous deterministic algo…
View article: Beyond the Lovász Local Lemma: Point to Set Correlations and Their Algorithmic Applications
Beyond the Lovász Local Lemma: Point to Set Correlations and Their Algorithmic Applications Open
Following the groundbreaking algorithm of Moser and Tardos for the Lovasz Local Lemma (LLL), there has been a plethora of results analyzing local search algorithms for various constraint satisfaction problems. The algorithms considered fal…
View article: Fisher Zeros and Correlation Decay in the Ising Model
Fisher Zeros and Correlation Decay in the Ising Model Open
The Ising model originated in statistical physics as a means of studying phase transitions in magnets, and has been the object of intensive study for almost a century. Combinatorially, it can be viewed as a natural distribution over cuts i…
View article: A New Perspective on Stochastic Local Search and the Lovasz Local Lemma.
A New Perspective on Stochastic Local Search and the Lovasz Local Lemma. Open
We present a new perspective on the analysis of stochastic local search algorithms via linear algebra. Our key insight is that LLL-inspired convergence arguments can be seen as a method for bounding the spectral radius of a matrix specifyi…
View article: Spatial Mixing and Non-local Markov chains
Spatial Mixing and Non-local Markov chains Open
We consider spin systems with nearest‐neighbor interactions on an n ‐vertex d ‐dimensional cube of the integer lattice graph . We study the effects that the strong spatial mixing condition (SSM) has on the rate of convergence to equilibriu…
View article: Spatial Mixing and Systematic Scan Markov chains
Spatial Mixing and Systematic Scan Markov chains Open
We consider spin systems on the integer lattice graph $\mathbb{Z}^d$ with nearest-neighbor interactions. We develop a combinatorial framework for establishing that exponential decay with distance of spin correlations, specifically the stro…
View article: Dynamics of lattice triangulations on thin rectangles
Dynamics of lattice triangulations on thin rectangles Open
We consider random lattice triangulations of $n\times k$ rectangular regions with weight $λ^{|σ|}$ where $λ>0$ is a parameter and $|σ|$ denotes the total edge length of the triangulation. When $λ\in(0,1)$ and $k$ is fixed, we prove a tight…