arXiv subjects

Remy Kassem

Publications and source records attributed to Remy Kassem.

5 recordsLinked to original sources

arXiv2026

Dispersive decay bounds for the SSH model on the half-line

We study the Schr\"odinger flow for the SSH model, a class of self-adjoint discrete dimer lattice Hamiltonians on the half-line. Using oscillatory integral techniques, we prove dispersive time-decay estimates, which quantify the spreading of energy throughout the lattice for a localized initial condition. Furthermore, we determine precise dependence of the constants in the decay rates on the parameters of the Hamiltonian. The analysis is complicated by the fact that as a consequence of the boundary condition, the expression for the propagator contains oscillatory integrals with nonintegrable singularities.

math-ph↗

arXiv2025

Dispersive Decay Estimates for periodic Jacobi operators on the half-line

We establish dispersive time-decay estimates for periodic Jacobi operators on the discrete half-line, $\N$. Specifically, we prove $t^{-1/2}$ decay in the weighted $\ell^\infty_{-1}$ norm for all such operators. For the global $\ell^1 \to \ell^\infty$ decay estimate, we show that $t^{-1/3}$ decay holds under a nondegeneracy condition on the discriminant. Alternatively, for any even period $q\geq2$, if the continuous spectrum consists of exactly $q$ disjoint intervals (bands), we obtain a $t^{-1/(q+1)}$ decay rate without any further assumptions.

math.SP↗

arXiv2021

Two-type annihilating systems on the complete and star graph

Red and blue particles are placed in equal proportion through-out either the complete or star graph and iteratively sampled to take simple random walk steps. Mutual annihilation occurs when particles with different colors meet. We compare the time it takes to extinguish every particle to the analogous time in the (simple to analyze) one-type setting. Additionally, we study the effect of asymmetric particle speeds.

math.PR↗

arXiv2019

SIR epidemics on evolving graphs

We consider evoSIR, a variant of the SIR model, on Erd\H os-Renyi random graphs in which susceptibles with an infected neighbor break that connection at rate $ρ$ and rewire to a randomly chosen individual. We compute the critical infection rate $λ_c$ and the probability of a large epidemic by showing that they are the same for the delSIR model in which $S-I$ connections are deleted instead of rewired. The final size of a large delSIR epidemic has a continuous transition. Simulations suggest that the final size of a large evoSIR epidemic is discontinuous at $λ_c$.

math.PR↗

arXiv2019

The Contact Process on Periodic Trees

A little over 25 years ago Pemantle pioneered the study of the contact process on trees, and showed that the critical values $λ_1$ and $λ_2$ for global and local survival were different. Here, we will consider the case of trees in which the degrees of vertices are periodic. We will compute bounds on $λ_1$ and $λ_2$ and for the corresponding critical values $λ_g$ and $λ_\ell$ for branching random walk. Much of what we find for period two $(a,b)$ trees was known to Pemantle. However, two significant new results give sharp asymptotics for the critical value $λ_2$ of $(1,n)$ trees and generalize that result to the $(a_1,\ldots, a_k, n)$ tree when $\max_i a_i \le n^{1-ε}$ and $a_1 \cdots a_k = n^b$. We also give results for $λ_g$ and $λ_\ell$ on $(a,b,c)$ trees. Since the values come from solving cubic equations, the explicit formulas are not pretty, but it is surprising that they depend only on $a+b+c$ and $abc$.

math.PR↗