Abstract The arboreal gas is the probability measure on (unrooted spanning) forests of a graph in which each forest is weighted by a factor $\beta &gt;0$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:math> per edge. It arises as the $q\to 0$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:math> limit of the $q$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML…

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 combinatorially flexible version of Pirogov--Sinai theory for the hard-core model of independent sets. Our results illustrate that the main conclusions of Pirogov--Sinai theory can be obtained in significantly greater generality than that of $\mathbb Z^{d}$. The main ingredients …

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 combining the contour representation of quantum spin systems of this type due to Borgs, Kotecký, and Ueltschi with the algorithmic framework developed by Helmuth, Perkins, and Regts, and Borgs et al.

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 this survey we introduce these models, discuss their origins and main features, some existing tools available for their study, recent probabilistic results, and relations to other well-studied probabilistic models. Along the way we discuss some of the many open questions that remain.

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 scheme exists. This motivates an average-case question: are there classes of instances for which polynomial-time approximation schemes exist? We investigate this question for the random field Ising model on graphs with maximum degree Δ . We establish the existence of fully polyno…