arXiv subjects

Jaehun Lee

Publications and source records attributed to Jaehun Lee.

26 records · Page 2Linked to original sources

arXiv2019

Equivalence of the grand canonical ensemble and the canonical ensemble on 1d-lattice systems

We consider a one-dimensional lattice system of unbounded, real-valued spins with arbitrary strong, quadratic, finite-range interaction. We show the equivalence of the grand canonical ensemble (gce) and the canonical ensemble (ce), in the sense of observables and correlations. A direct consequence is that the correlations of the ce decay exponentially plus a volume correction term. The volume correction term is uniform in the external field, the mean spin and scales optimally in the system size. This extends prior results of Cancrini & Martinelli for bounded discrete spins to unbounded continuous spins. The result is obtained by adapting Cancrini & Martinelli's method combined with authors' recent approach on continuous real-valued spin systems.

math.PR↗

arXiv2018

Delocalization and Limiting Spectral Distribution of Erdős-Rényi Graphs with Constant Expected Degree

We consider Erdős-Rényi graphs $G(n,p_n)$ with large constant expected degree $λ$ and $p_n=λ/n$. Bordenave and Lelarge (2010) showed that the infinite-volume limit, in the Benjamini-Schramm topology, is a Galton-Watson tree with offspring distribution Pois($λ$) and the mean spectrum at the root of this tree has unbounded support and corresponds to the limiting spectral distribution of $G(n,p_n)$ as $n\to\infty$. We show that if one weights the edges by $1/\sqrtλ$ and sends $λ\to\infty$, then the support mostly vanishes and in fact, the limiting spectral distributions converge weakly to a semicircle distribution. We also find that for large $λ$, there is an orthonormal eigenvector basis of $G(n,p_n)$ such that most of the vectors delocalize with respect to the infinity norm, as $n\to\infty$. Our delocalization result provides a variant on a result of Tran, Vu and Wang (2013).

math.PR↗

arXiv2018

Heat kernel estimates for symmetric jump processes with mixed polynomial growths

In this paper, we study the transition densities of pure-jump symmetric Markov processes in $ {\mathbb R}^d$, whose jumping kernels are comparable to radially symmetric functions with mixed polynomial growths. Under some mild assumptions on their scale functions, we establish sharp two-sided estimates of transition densities (heat kernel estimates) for such processes. This is the first study on global heat kernel estimates of jump processes (including non-Lévy processes) whose weak scaling index is not necessarily strictly less than 2. As an application, we proved that the finite second moment condition on such symmetric Markov process is equivalent to the Khintchine-type law of iterated logarithm at the infinity.

math.PR↗

arXiv2018

Boundary regularity for nonlocal operators with kernels of variable orders

We study the boundary regularity of solutions of the Dirichlet problem for the nonlocal operator with a kernel of variable orders. Since the order of differentiability of the kernel is not represented by a single number, we consider the generalized Hölder space. We prove that there exists a unique viscosity solution of $Lu = f$ in $D$, $u=0$ in $\mathbb{R}^n \setminus D$, where $D$ is a bounded $C^{1,1}$ open set, and that the solution $u$ satisfies $u \in C^V(D)$ and $u/V(d_D) \in C^α(D)$ with the uniform estimates, where $V$ is the renewal function and $d_D(x) = \mbox{dist}(x, \partial D)$.

math.AP↗

arXiv2018

Zeros of random polynomials and its higher derivatives

In this article we study the limiting empirical measure of zeros of higher derivatives for sequences of random polynomials. We show that these measures agree with the limiting empirical measure of zeros of corresponding random polynomials. Various models of random polynomials are considered by introducing randomness through multiplying a factor with a random zero or removing a zero at random for a given sequence of deterministic polynomials. We also obtain similar results for random polynomials whose zeros are given by i.i.d. random variables. As an application, we show that these phenomenon appear for random polynomials whose zeros are given by the 2D Coulomb gas density.

math.PR↗

arXiv2018

Heat kernels of non-symmetric jump processes with exponentially decaying jumping kernel

In this paper we study the transition densities for a large class of non-symmetric Markov processes whose jumping kernels decay exponentially or subexponentially. We obtain their upper bounds which also decay at the same rate as their jumping kernels. When the lower bounds of jumping kernels satisfy the weak upper scaling condition at zero, we also establish lower bounds for the transition densities, which are sharp.

math.PR↗

arXiv2015

Action-angle variables in curved space-time

We construct a relativistic and curved space version of action- angle variables for a particle trapped in a gravity and electromagnetic background with time-like isometry. As an example, we consider a particle in AdS background. Furthermore, we obtain the semiclassical quantisation of its energy levels.

gr-qc↗

arXiv2014

On Isomorphism Classes of Generalized Fibonacci Cubes

The generalized Fibonacci cube $Q_d(f)$ is the subgraph of the $d$-cube $Q_d$ induced on the set of all strings of length $d$ that do not contain $f$ as a substring. It is proved that if $Q_d(f) \cong Q_d(f')$ then $|f|=|f'|$. The key tool to prove this result is a result of Guibas and Odlyzko about the autocorrelation polynomial associated to a binary string. It is also proved that there exist pairs of strings $f, f'$ such that $Q_d(f) \cong Q_d(f')$, where $|f| \ge \frac{2}{3}(d+1)$ and $f'$ cannot be obtained from $f$ by its reversal or binary complementation. Strings $f$ and $f'$ with $|f|=|f'|=d-1$ for which $Q_d(f) \cong Q_d(f')$ are characterized.

math.CO↗