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Recently it has been claimed that ordinary perturbation theory (OPT) gives incorrect weak coupling expansions for lattice O(N) non-linear sigma models in the infinite volume limit, and in particular that the two-dimensional non-abelian…
Several years ago it was found that perturbation theory for two-dimensional O(N) models depends on boundary conditions even after the infinite volume limit has been taken termwise, provided $N>2$. There ensued a discussion whether the…
It is shown that perturbation theory in $2D$ nonlinear $\sigma$-models as well gauge theories in dimension $D\geq 2$ produces answers that depend on boundary conditions even after the infinite volume limit has been taken. This unphysical…
We discuss some aspects of the continuum limit of some lattice models, in particular the $2D$ $O(N)$ models. The continuum limit is taken either in an infinite volume or in a box whose size is a fixed fraction of the infinite volume…
Noncompact SO(1,N) sigma-models are studied in terms of their large N expansion in a lattice formulation in dimensions d \geq 2. Explicit results for the spin and current two-point functions as well as for the Binder cumulant are presented…
Nonlinear sigma models with non-compact target space and non-amen-able symmetry group were introduced long ago in the study of disordered electron systems. They also occur in dimensionally reduced quantum gravity; recently they have been…
The topological susceptibility of $2d$ $\mathrm{CP}^{N-1}$ models is expected, based on perturbative computations, to develop a divergence in the limit $N \to 2$, where these models reduce to the well-known non-linear $\mathrm{O}(3)$…
We consider the Laplacian in a tubular neighbourhood of a hyperplane subjected to non-self-adjoint $\mathcal{PT}$-symmetric Robin boundary conditions. Its spectrum is found to be purely essential and real for constant boundary conditions.…
We study field theories in the limit that a compactified dimension becomes lightlike. In almost all cases the amplitudes at each order of perturbation theory diverge in the limit, due to strong interactions among the longitudinal zero…
We analyze the arguments put forward recently by Niedermayer et al in favor of the correctness of conventional perturbation theory in non-Abelian models and supposedly showing that our super-instanton counterexample was sick. We point out…
We study a simple model for which perturbation theory gives ultravioletly divergent results. We show that when the eigen-solution problem of the Hamiltonian of the model is treated nonperturbatively, it is possible for eigenenergies of the…
Built upon the proposal of Kaplan et.al. [hep-lat/0206109], we construct noncommutative lattice gauge theory with manifest supersymmetry. We show that such theory is naturally implementable via orbifold conditions generalizing those used by…
Supersymmetry is a prominent candidate for physics beyond the standard model. In order to compute the spectrum of supersymmetric theories, we employ nonperturbative lattice QFT techniques which due to the discretisation of spacetime violate…
We show that the spacing between eigenvalues of the discrete 1D Hamiltonian with arbitrary potentials which are bounded, and with Dirichlet or Neumann Boundary Conditions is bounded away from zero. We prove an explicit lower bound, given by…
This is the third of the series of articles on the large-$N$ two-dimensional $\mathbb{CP}^{N-1}$ sigma model, defined on a finite space interval $L$ with Dirichlet boundary conditions. Here the cases of the general Dirichlet boundary…
We provide upper bounds on the perturbation of invariant subspaces of normal matrices measured using a metric on the space of vector subspaces of $\mathbb{C}^n$ in terms of the spectrum of both the unperturbed \& perturbed matrices, as well…
For a fixed symmetric matrix A and symmetric perturbation E we develop purely deterministic bounds on how invariant subspaces of A and A+E can differ when measured by a suitable "row-wise" metric rather than via traditional measures of…
We study cutoff and lattice effects in the O(n) symmetric $\phi^4$ theory for a $d$-dimensional cubic geometry of size $L$ with periodic boundary conditions. In the large-N limit above $T_c$, we show that $\phi^4$ field theory at finite…
2D nonlinear sigma models with Hermitian symmetric target admit a theta-term, which couples the field theory to the topological charge of its instanton gas. At the special coupling theta = pi, by what is nowadays attributed to a…
Biggins [Uniform convergence of martingales in the branching random walk. {\em Ann. Probab.}, 20(1):137--151, 1992] proved local uniform convergence of additive martingales in $d$-dimensional supercritical branching random walks at complex…