Robert B. Israel

Finding Generators for Markov Chains via Empirical Transition Matrices, with Applications to Credit Ratings

In this paper we identify conditions under which a true generator does or does not exist for an empirically observed Markov transition matrix. We show how to search for valid generators and choose the “correct” one that is the most compatible with bond rating behaviors. We also show how to obtain an approximate generator when a true generator does not exist. We give illustrations using credit rating transition matrices published by Moodys and by Standard and Poors.

Phase Transitions and Reflection Positivity. I. General Theory and Long Range Lattice Models

We systematize the study of reflection positivity in statistical mechanical models, and thereby two techniques in the theory of phase transitions: the method of infrared bounds and the chessboard method of estimating contour probabilities in Peierls arguments. We illustrate the ideas by applying them to models with long range interactions in one and two dimensions. Additional applications are discussed in a second paper.

High-temperature analyticity in classical lattice systems

We prove analyticity of the correlation functions for classical lattice systems, including “continuous-spin” systems, at high temperatures and in strong external fields. For systems whose configuration spaces are homogeneous spaces for compact groups (e.g. Ising, plane rotator and classical Heisenberg models), improved estimates on the region of analyticity are obtained by generalizing an integral equation of Gruber and Merlini. Exponential cluster properties are also obtained for such systems with a finite-range interaction.

Phase Transitions and Reflection Positivity. II. Lattice Systems with Short-Range and Coulomb Interactions

This is the second paper in a series descr ibing appl ica t ions of reflection posi t iv i ty (RP) to proving the existence of phase t ransi t ions in mode l systems. We exploi t : (1) the Peierls chessboard a rgument first used by G l i m m et al. ~22~ and fur ther developed by Fr6h l ich and Lieb~la~; (2) the me thod o f infrared bounds , first used by Fr6h l i ch et al. ~17~ and extended to qua n tum systems by Dyson et aI56~ Reviews of some of these ideas can be found in Refs. 8, 9, 18, 30, 36, and 49. In pape r I ~zl~ of this series, ~ we presented a general f ramework, and in a th i rd paper , a2~ we give appl ica t ions to q u a n t u m field theories. In this paper , we deal with shor t range lat t ice models and C o u l o m b lat t ice gases. In I, we discussed long-range lat t ice models . A fur ther app l ica t ion to a mode l of a l iquid crystal has been found by He i lmann and Lieb ~26~ and one to d ipole lat t ice gases by Fr6h l ich and SpencerJ 19~

Discrete optimization using quantum annealing on sparse Ising models

This paper discusses techniques for solving discrete optimization problems using quantum annealing. Practical issues likely to affect the computation include precision limitations, finite temperature, bounded energy range, sparse connectivity, and small numbers of qubits. To address these concerns we propose a way of finding energy representations with large classical gaps between ground and first excited states, efficient algorithms for mapping non-compatible Ising models into the hardware, and the use of decomposition methods for problems that are too large to fit in hardware. We validate the approach by describing experiments with D-Wave quantum hardware for low density parity check decoding with up to 1000 variables.

Existence of phase transitions for long-range interactions

Using a theorem about tangent functionals to convex functions, we obtain existence results for phase transitions. In a “large” Banach space of interactions very pathological behavior is found. In spaces of more “reasonable” interactions we obtain co-existing phases which differ in the expectation of a given observable, as well as broken translation invariance due to long-range order. As an example we consider the isotropic Heisenberg model.

Quark confinement in the two-dimensional lattice Higgs-Villain model

We prove quark confinement in the two-dimensional lattice Higgs-Villain model in the weak coupling region by using a Kirkwood-Salsburg equations for unbounded spins.

Stronger Players Need Not Win More Knockout Tournaments

Abstract We present a counterexample to the following conjecture of F.R.K. Chung and F.K. Hwang (1978): for any knockout tournament plan and any preference scheme satisfying strong stochastic transitivity, if all assignments of players to starting positions are equally likely, then stronger players (as defined by the preference scheme) are more likely to win the tournament.

Mapping Constrained Optimization Problems to Quantum Annealing with Application to Fault Diagnosis

Current quantum annealing (QA) hardware suffers from practical limitations such as finite temperature, sparse connectivity, small qubit numbers, and control error. We propose new algorithms for mapping boolean constraint satisfaction problems (CSPs) onto QA hardware mitigating these limitations. In particular we develop a new embedding algorithm for mapping a CSP onto a hardware Ising model with a fixed sparse set of interactions, and propose two new decomposition algorithms for solving problems too large to map directly into hardware. The mapping technique is locally-structured, as hardware compatible Ising models are generated for each problem constraint, and variables appearing in different constraints are chained together using ferromagnetic couplings. In contrast, global embedding techniques generate a hardware independent Ising model for all the constraints, and then use a minor-embedding algorithm to generate a hardware compatible Ising model. We give an example of a class of CSPs for which the scaling performance of the D-Wave hardware using the local mapping technique is significantly better than global embedding. We validate the approach by applying D-Waves QA hardware to circuit-based fault-diagnosis. For circuits that embed directly, we find that the hardware is typically able to find \emph{all} solutions from a min-fault diagnosis set of size N using 1000N samples, using an annealing rate that is 25 times faster than a leading SAT-based sampling method. Further, we apply decomposition algorithms to find min-cardinality faults for circuits that are up to 5 times larger than can be solved directly on current hardware.

Generic triviality of phase diagrams in spaces of long-range interactions

We show that interactions with multiple translation-invariant equilibrium states form a very “thin” set in spaces of long-range interactions of classical or quantum lattice systems. For example, generic finite-dimensional subspaces do not intersect this set. This constitutes a severe violation of the Gibbs Phase Rule.