Narrow Results

Short distance scaling limits of a class of integrable models on two-dimensional Minkowski space are considered in the algebraic framework of quantum field theory. Making use of the wedge-local quantum fields generating these models, it is shown that massless scaling limit theories exist, and decompose into (twisted) tensor products of chiral, translation-dilation covariant field theories. On the subspace which is generated from the vacuum by the observables localized in finite light ray intervals, this symmetry can be extended to the Möbius group. The structure of the interval-localized algebras in the chiral models is discussed in two explicit examples.

We consider Vlasov-type scaling for the Glauber dynamics in continuum with a positive integrable potential, and construct rescaled and limiting evolutions of correlation functions. Convergence to the limiting evolution for the positive density system in infinite volume is shown. Chaos preservation property of this evolution gives a possibility to derive a nonlinear Vlasov-type equation for the particle density of the limiting system.

We consider a linear equation ${\partial }_{t}u={ℒ}u$, where ${ℒ}$ is a generator of a semigroup of linear operators on a certain Hilbert space related to an initial condition $u(0)$ being a generalised stationary random field on ${ℝ}^d$. We show the existence and uniqueness of generalised solutions to such initial value problems. Then we investigate their scaling limits.

A trace formula for Toeplitz operators was proved by Boutet de Monvel and Guillemin in the setting of general Toeplitz structures. Here, we give a local version of this result for a class of Toeplitz operators related to continuous groups of symmetries on quantizable compact symplectic manifolds. The local trace formula involves certain scaling asymptotics along the clean fixed locus of the Hamiltonian flow of the symbol, reminiscent of the scaling asymptotics of the equivariant components of the Szegö kernel along the diagonal.

We consider the eigenvalues of symplectic elliptic Ginibre matrices which are known to form a Pfaffian point process whose correlation kernel can be expressed in terms of the skew-orthogonal Hermite polynomials. We derive the scaling limits and the convergence rates of the correlation functions at the real bulk/edge of the spectrum, which in particular establishes the local universality at strong non-Hermiticity. Furthermore, we obtain the subleading corrections of the edge correlation kernels, which depend on the non-Hermiticity parameter contrary to the universal leading term. Our proofs are based on the asymptotic behavior of the complex elliptic Ginibre ensemble due to Lee and Riser as well as on a version of the Christoffel–Darboux identity, a differential equation satisfied by the skew-orthogonal polynomial kernel.

In this paper, we consider the elliptic Ginibre matrices in the orthogonal symmetry class that interpolates between the real Ginibre ensemble and the Gaussian orthogonal ensemble. We obtain the finite size corrections of the real eigenvalue densities in both the global and edge scaling regimes, as well as in both the strong and weak non-Hermiticity regimes. Our results extend and provide the rate of convergence to the previous recent findings in the aforementioned limits. In particular, in the Hermitian limit, our results recover the finite size corrections of the Gaussian orthogonal ensemble established by Forrester, Frankel and Garoni.

Many packaging experts believe Moore’s law will hold for another decade due to recent technological efforts to merge multiple integrated circuit dies horizontally and vertically into a single package, commonly referred to as heterogeneous integration. With simultaneous shrinkage in chip size and larger heat dissipation, the flux values have increased exponentially. To tackle the skyrocketing thermal management challenges, the electronics industry and academic research community have examined various thermal management solutions. Embedded cooling eliminates most of the sequential conduction resistances from the chip to ambient, unlike the separable cold plates/heat sinks. Though there is a high heat transfer potential by embedding the cooling solution onto an electronic chip, there are still technological risks associated with the implementation of these technologies and uncertainty about which technologies should be adopted. This chapter discusses the thermal spreading limits, fluid selection considerations, as well as conventional and additive manufacturing-based embedded-cooling technologies.

We discuss asymptotics for large random planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with index α ∈ (1,2). We view the vertex-set of a map as a metric space by endowing it with the usual graph distance. When the number n of vertices of the map tends to infinity, this metric space, rescaled by the factor n^-1/2α, converges in distribution as n → ∞, at least along suitable subsequences, towards a limiting random compact metric space whose Hausdorff dimension is equal to 2α.

This is a short presentation of the article [1], to which the interested reader is referred for more details.

We review recent results with D. Chelkak and K. Izyurov [10], where we rigorously prove existence and conformal invariance of scaling limits of magnetization and multi-point spin correlations in the critical Ising model on an arbitrary simply connected planar domain. This solves a number of conjectures coming from physical and mathematical literatures. The proof is based on convergence results for discrete holomorphic spinor observables.