Tyler Helmuth
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View article: Pirogov--Sinai Theory Beyond Lattices
Pirogov--Sinai Theory Beyond Lattices Open
Pirogov--Sinai theory is a well-developed method for understanding the low-temperature phase diagram of statistical mechanics models on lattices. Motivated by physical and algorithmic questions beyond the setting of lattices, we develop a …
View article: Percolation transition for random forests in $d\geqslant 3$
Percolation transition for random forests in $d\geqslant 3$ Open
The arboreal gas is the probability measure on (unrooted spanning) forests of a graph in which each forest is weighted by a factor $\beta >0$ per edge. It arises as the $q\to 0$ limit of the $q$ -state random cluster model with $…
View article: Spin systems with hyperbolic symmetry: a survey
Spin systems with hyperbolic symmetry: a survey Open
Spin systems with hyperbolic symmetry originated as simplified models for the Anderson metal–insulator transition, and were subsequently found to exactly describe probabilistic models of linearly reinforced walks and random forests. In thi…
View article: Efficient Algorithms for Approximating Quantum Partition Functions at Low Temperature
Efficient Algorithms for Approximating Quantum Partition Functions at Low Temperature Open
We establish an efficient approximation algorithm for the partition functions of a class of quantum spin systems at low temperature, which can be viewed as stable quantum perturbations of classical spin systems. Our algorithm is based on c…
View article: Approximation Algorithms for the Random Field Ising Model
Approximation Algorithms for the Random Field Ising Model Open
Approximating the partition function of the ferromagnetic Ising model with general external fields is known to be #BIS-hard in the worst case, even for bounded-degree graphs, and it is widely believed that no polynomial-time approximation …
View article: Percolation on hypergraphs and the hard-core model
Percolation on hypergraphs and the hard-core model Open
We prove tight bounds on the site percolation threshold for $k$-uniform hypergraphs of maximum degree $Δ$ and for $k$-uniform hypergraphs of maximum degree $Δ$ in which any pair of edges overlaps in at most $r$ vertices. The hypergraphs th…
View article: Directed Spatial Permutations on Asymmetric Tori
Directed Spatial Permutations on Asymmetric Tori Open
We investigate a model of random spatial permutations on two-dimensional tori, and establish that the joint distribution of large cycles is asymptotically given by the Poisson--Dirichlet distribution with parameter one. The asymmetry of th…
View article: Efficient sampling and counting algorithms for the Potts model on <i>ℤ</i><sup><i>d</i></sup> at all temperatures
Efficient sampling and counting algorithms for the Potts model on <i>ℤ</i><sup><i>d</i></sup> at all temperatures Open
For and all we give an efficient algorithm to approximately sample from the ‐state ferromagnetic Potts and random cluster models on finite tori for any inverse temperature . This shows that the physical phase transition of the Potts model …
View article: Correlation decay for hard spheres via Markov chains
Correlation decay for hard spheres via Markov chains Open
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View article: Spin systems with hyperbolic symmetry: a survey
Spin systems with hyperbolic symmetry: a survey Open
Spin systems with hyperbolic symmetry originated as simplified models for the Anderson metal--insulator transition, and were subsequently found to exactly describe probabilistic models of linearly reinforced walks and random forests. In th…
View article: Approximation algorithms for the random-field Ising model
Approximation algorithms for the random-field Ising model Open
Approximating the partition function of the ferromagnetic Ising model with general external fields is known to be #BIS-hard in the worst case, even for bounded-degree graphs, and it is widely believed that no polynomial-time approximation …
View article: Percolation transition for random forests in $d\geq 3$
Percolation transition for random forests in $d\geq 3$ Open
The arboreal gas is the probability measure on (unrooted spanning) forests of a graph in which each forest is weighted by a factor $β>0$ per edge. It arises as the $q\to 0$ limit of the $q$-state random cluster model with $p=βq$. We prove …
View article: Efficient algorithms for approximating quantum partition functions
Efficient algorithms for approximating quantum partition functions Open
We establish a polynomial-time approximation algorithm for partition functions of quantum spin models at high temperature. Our algorithm is based on the quantum cluster expansion of Netočný and Redig and the cluster expansion approach to d…
View article: Finite-size scaling, phase coexistence, and algorithms for the random cluster model on random graphs
Finite-size scaling, phase coexistence, and algorithms for the random cluster model on random graphs Open
For $Δ\ge 5$ and $q$ large as a function of $Δ$, we give a detailed picture of the phase transition of the random cluster model on random $Δ$-regular graphs. In particular, we determine the limiting distribution of the weights of the order…
View article: Efficient sampling and counting algorithms for the Potts model on ℤᵈ at all temperatures
Efficient sampling and counting algorithms for the Potts model on ℤᵈ at all temperatures Open
For d ≥ 2 and all q≥ q 0(d) we give an efficient algorithm to approximately sample from the q-state ferromagnetic Potts and random cluster models on the torus (ℤ / n ℤ ) d for any inverse temperature β≥ 0. This stands in contrast to Markov…
View article: Correlation decay for hard spheres via Markov chains
Correlation decay for hard spheres via Markov chains Open
We improve upon all known lower bounds on the critical fugacity and critical density of the hard sphere model in dimensions two and higher. As the dimension tends to infinity our improvements are by factors of $2$ and $1.7$, respectively. …
View article: Efficient sampling and counting algorithms for the Potts model on $\mathbb Z^d$ at all temperatures
Efficient sampling and counting algorithms for the Potts model on $\mathbb Z^d$ at all temperatures Open
For $d \ge 2$ and all $q\geq q_{0}(d)$ we give an efficient algorithm to approximately sample from the $q$-state ferromagnetic Potts and random cluster models on finite tori $(\mathbb Z / n \mathbb Z )^d$ for any inverse temperature $β\geq…
View article: Efficient sampling and counting algorithms for the Potts model on\n $\\mathbb Z^d$ at all temperatures
Efficient sampling and counting algorithms for the Potts model on\n $\\mathbb Z^d$ at all temperatures Open
For $d \\ge 2$ and all $q\\geq q_{0}(d)$ we give an efficient algorithm to\napproximately sample from the $q$-state ferromagnetic Potts and random cluster\nmodels on finite tori $(\\mathbb Z / n \\mathbb Z )^d$ for any inverse\ntemperature…
View article: Algorithmic Pirogov-Sinai theory
Algorithmic Pirogov-Sinai theory Open
We develop an efficient algorithmic approach for approximate counting and sampling in the low-temperature regime of a broad class of statistical physics models on finite subsets of the lattice $\mathbb Z^d$ and on the torus $(\mathbb Z/n \…
View article: The continuous-time lace expansion
The continuous-time lace expansion Open
We derive a continuous-time lace expansion for a broad class of self-interacting continuous-time random walks. Our expansion applies when the self-interaction is a sufficiently nice function of the local time of a continuous-time random wa…
View article: Dynkin isomorphism and Mermin–Wagner theorems for hyperbolic sigma models and recurrence of the two-dimensional vertex-reinforced jump process
Dynkin isomorphism and Mermin–Wagner theorems for hyperbolic sigma models and recurrence of the two-dimensional vertex-reinforced jump process Open
We prove the vertex-reinforced jump process (VRJP) is recurrent in two dimensions for any translation invariant finite-range initial rates. Our proof has two main ingredients. The first is a direct connection between the VRJP and sigma mod…
View article: Loop-weighted walk
Loop-weighted walk Open
Loop-weighted walk with parameter \lambda \geq 0 is a non-Markovian model of random walks that is related to the loop O(N) model of statistical mechanics. A walk receives weight \lambda^{k} if it contains k loops; whether this is a reward …