Mathematical Physics
This paper considers several aspects of random matrix universality in deep neural networks. Motivated by recent experimental work, we use universal properties of random matrices related to local statistics to derive practical implications…
We examine the asymptotics of the moments of characteristic polynomials of $N\times N$ matrices drawn from the Hermitian ensembles of Random Matrix Theory, in the limit as $N\to\infty$. We focus in particular on the Gaussian Unitary…
Representation theory and the theory of symmetric functions have played a central role in Random Matrix Theory in the computation of quantities such as joint moments of traces and joint moments of characteristic polynomials of matrices…
In quenched disordered systems, the existence of ordering is generally believed to be only possible in the weak disorder regime (disregarding models of spin-glass type). In particular, sufficiently large random fields is expected to…
In this paper, we introduce the $q$-Heisenberg algebra in the tensor product space $\otimes^2$. We establish its algebraic properties and provide applications to the theory of non-monogenic functions. Our results extend known constructions…
Enhanced binding of a quantum particle coupled to a quantized field means that the Hamiltonian of the particle alone does not have a bound state, while the particle-field Hamiltonian does. For the Pauli--Fierz model, this is usually shown…
In this paper, we consider the resonance problem for the cubic nonlinear Helmholtz equation in the subwavelength regime. We derive a discrete model for approximating the subwavelength resonances of finite systems of high-contrast resonators…
The study of the Two-Body and Circular Restricted Three-Body Problems in the field of aerospace engineering and sciences is deeply important because they help describe the motion of both celestial and artificial satellites. With the growing…
On real metric manifolds admitting a co-dimension one foliation, sectorial operators are introduced that interpolate between the generalized Laplacian and the d'Alembertian. This is used to construct a one-parameter family of analytic…
A GL$(n)$ quantum integrable system generalizing the asymmetric five vertex spin chain is shown to encode the ring relations of the equivariant quantum cohomology and equivariant quantum K-theory ring of flag varieties. We also show that…
The fermionic von Neumann entropy, the fermionic entanglement entropy and the fermionic relative entropy are defined for causal fermion systems. Our definition makes use of entropy formulas for quasi-free fermionic states in terms of the…
We construct sub-grid scale models of incompressible fluids by considering expectations of semi-martingale Lagrangian particle trajectories. Our construction is based on the Lagrangian decomposition of flow maps into mean and fluctuation…
In this paper we introduce the $q$-deformed Heisenberg picture equation. We consider some examples such as : the spinless particle, the electr\'on interaction with a magnnetic field and $q$-deformed harmonnic oscillator. The $q$-Heisenberg…
We present several known solutions to the two-dimensional Ising model. This review originated from the ``Ising 100'' seminar series held at Bo\u{g}azi\c{c}i University, Istanbul, in 2024.
This paper analyzes the scattering matrix for two unbounded self-adjoint operators: the standard Laplace operator in three-dimensional space and a second operator that differs from the first by an infinite sum of zero-range potentials.
Hodograph equations for the Euler equation in curved spaces with constant pressure are discussed. It is shown that the use of known results concerning geodesics and associated integrals allows to construct several types of hodograph…
Euler derived the differential equations of elastica by the variational method in 1744, but his original derivation has never been properly interpreted or explained in terms of modern mathematics. We elaborate Euler's original derivation of…
In this paper, we study Hamiltonian operators which are sum of a first order operator and of a Poisson tensor, in two spatial independent variables. In particular, a complete classification of these operators is presented in two and three…
The Thouless theory of quantum pumps establishes the conditions for quantized particle transport per cycle, and determines its value. When describing the pump from a moving reference frame, transported and existing charges transform, though…
This paper presents the axioms for a quantum Yang-Mills theory in the Minkowski spacetime. There are two routes of analytic continuation for the Schwinger functions, namely the Wightman functions and time-ordered products of field…