arXiv subjects
Publications and source records attributed to Ron Nissim.
We make rigorous the physics prediction that lattice Yang-Mills theories with gauge groups which have trivial centers do not satisfy Wilson's criterion for quark confinement. Specifically we prove that $\mathrm{SO}(3)$ lattice Yang-Mills theory does not satisfy Wilson's criterion in a strong coupling regime.
A foundational result in constructive quantum field theory is the construction of the free bosonic quantum field theory in four-dimensional Euclidean spacetime and the proof that it satisfies the Glimm-Jaffe axioms, a variant of the Osterwalder-Schrader axioms. We present a formalization of this result in the Lean 4 interactive theorem prover. The project is intended as a proof of concept that extended arguments in mathematical physics can be translated into machine-checked proofs using existing AI tools. We begin by introducing interactive theorem proving and constructive quantum field theory, then describe our formalization and the design decisions that shaped it. We also explain the methods we used, including coding assistants, and conclude by considering how AI assisted formalization may influence the future of theoretical physics. Our original release assumed three results, Minlos' theorem, the nuclear property of Schwartz space, and Goursat's theorem. In subsequent releases from our group and from contributors from the Lean community, these assumptions have been proven (or avoided), so that the OS/GJ axioms are now proven using only Lean and its library Mathlib.
We establish a mass gap, prove the existence of a unique infinite volume limit, and give a new proof of the large $N$ limit for $\mathrm{U}(N)$ lattice Yang-Mills theory in the 't Hooft regime. These results were previously obtained for $\mathrm{SU}(N)$ and $\mathrm{SO}(N)$ lattice Yang-Mills theories as applications of the mixing of the associated Langevin dynamics, which is verified via the Bakry-\'Emery criterion [SZZ23]. For $\mathrm{U}(N)$, however, this approach fails because its Ricci curvature is not uniformly positive, and as a result the Bakry-\'Emery condition cannot be easily verified. To overcome this obstacle, we recast the $\mathrm{U}(N)$ theory as a random-environment $\mathrm{SU}(N)$ model, where the randomness arises from a $\mathrm{U}(1)$ field, and combine cluster-expansion and Langevin-dynamics techniques to analyze the resulting $\mathrm{U}(1)\times\mathrm{SU}(N)$ model.
In this note, we observe that the dynamical approach to lattice Yang-Mills set forth in [SZZ23] may also be applied to prove Wilson's area law in the 't Hooft regime of parameters. The main point is to verify the mass gap condition from [DF80], from which area law directly follows. Our results apply for gauge groups $G \in \{\mathrm{U}(N), \mathrm{SU}(N), \mathrm{SO}(2N)\}$, which all have nontrivial center (which is one of the key assumptions in [DF80]).
We extend the parameter regimes for which area law is proven for pure $\mathrm{U}(N)$ lattice Yang-Mills theories, in particular when $N$ is large. This improves on a classical result of Osterwalder-Seiler from 1978. To do so, we view the master loop equation as a linear inhomogeneous equation for Wilson string expectations, and then prove an a priori bound for solutions to the equation. The main novelty is in how we deal with the merger term in the master loop equation. This is done by introducing a truncated model for which the merger term is unproblematic, and then showing that the truncated model well approximates the original model.
Weingarten functions provide a tool for computing Haar measure matrix integrals of polynomials in the matrix entries. An important property of Weingarten functions, is their particularly simple large $N$ limits. In 2017 Benoit Collins and Sho Matsumoto studied when this limit holds for Weingarten functions associated to integrals of products of $2n$ matrix entries, as $n \to \infty$, together with the matrix size $N$. They showed that the large $N$ limit is uniformly achieved as long as $n=o(N^{4/7})$, a result which already has applications to strong asymptotic freeness. However, their result is not optimal. They conjectured that their result should actually hold up to $n=o(N^{2/3})$ which is optimal. We prove this conjecture for the matrix groups $G \in \{\mathrm{U}(N)$, $\mathrm{O}(N)$, $\mathrm{Sp}(N)\}$. The proof proceeds by introducing a Markov process on permutations (pairings) which we call the unitary (orthogonal) $\textit{Weingarten process}$. We believe this process may have further applications to the theory of Weingarten functions. We also prove two new bounds regarding the large $N$ limit of the Weingarten function in the regimes when $n=o(N^{4/5})$, and $n=o(N)$.
In previous work, a description of the result of applying the Householder tridiagonalization algorithm to a G$\beta$E random matrix is provided by Edelman and Dumitriu. The resulting tridiagonal ensemble makes sense for all $\beta>0$, and has spectrum given by the $\beta$-ensemble for all $\beta>0$. Moreover, the tridiagonal model has useful stochastic operator limits which was introduced and analyzed in subsequent studies. In this work, we analogously study the result of applying the Householder tridiagonalization algorithm to a G$\beta$E process which has eigenvalues governed by $\beta$-Dyson Brownian motion. We propose an explicit limit of the upper left $k \times k$ minor of the $n \times n$ tridiagonal process as $n \to \infty$ and $k$ remains fixed. We prove the result for $\beta=1$, and also provide numerical evidence for $\beta=1,2,4$. This leads us to conjecture the form of a dynamical $\beta$-stochastic Airy operator with smallest $k$ eigenvalues evolving according to the $n \to \infty$ limit of the largest, centered and re-scaled, $k$ eigenvalues of $\beta$-Dyson Brownian motion.
In this paper we provide a geometric derivation of the master loop equation for the lattice Yang-Mills model with structure group $G \in \{SO(N),SU(N), U(N)\}$. This approach is based on integration by parts on $G$. In the appendix we compare our approach to that of \cite{Ch19a} and \cite{J16} based on Schwinger-Dyson equations, and \cite{SheSmZh22} based on stochastic analysis. In particular these approaches are all easily seen to be equivalent. The novelty in our approach is the use of intrinsic geometry of $G$ which we believe simplifies the derivation.
We consider the Kardar-Parisi-Zhang (KPZ) fixed point $\mathrm{H}(x,\tau)$ with the narrow-wedge initial condition and investigate the distribution of $\mathrm{H}(x,\tau)$ conditioned on a large height at an earlier space-time point $\mathrm{H}(x',\tau')$. As $\mathrm{H}(x',\tau')$ tends to infinity, we prove that the conditional one-point distribution of $\mathrm{H}(x,\tau)$ in the regime $\tau>\tau'$ converges to the Gaussian Unitary Ensemble (GUE) Tracy-Widom distribution and that the next two lower-order error terms can be expressed as derivatives of the Tracy-Widom distribution. The lowe order expansion here is analogue to the Edgeworth expansion in the central limit theorem. These KPZ-type limiting behaviors are different from the Gaussian-type ones obtained in \cite{Liu-Wang22} where they study the finite-dimensional distribution of $\mathrm{H}(x,\tau)$ conditioned on a large height at a later space-time point $\mathrm{H}(x',\tau')$. They show, with the narrow-wedge initial condition, that the conditional random field $\mathrm{H}(x,\tau)$ in the regime $\tau<\tau'$ converges to the minimum of two independent Brownian bridges modified by linear drifts as $\mathrm{H}(x',\tau')$ goes to infinity. The two results stated above provide the phase diagram of the asymptotic behaviors of a conditional law of KPZ fixed point in the regimes $\tau>\tau'$ and $\tau<\tau'$ when $\mathrm{H}(x',\tau')$ goes to infinity.