arXiv subjects

Takashi Kumagai

Publications and source records attributed to Takashi Kumagai.

At least 37 records · Page 2Linked to original sources

arXiv2022

Limit theorems for some long range random walks on torsion free nilpotent groups

We consider a natural class of long range random walks on torsion free nilpotent groups and develop limit theorems for these walks. Given the original discrete group $Γ$ and a random walk $(S_n)_ {n\ge1}$ driven by a certain type of symmetric probability measure $μ$, we construct a homogeneous nilpotent Lie group $G_\bullet(Γ,μ)$ which carries an adapted dilation structure and a stable-like process $(X_t)_{ t\ge0}$ which appears in a Donsker-type functional limit theorem as the limit of a rescaled version of the random walk. Both the limit group and the limit process on that group depend on the measure $μ$. In addition, the functional limit theorem is complemented by a local limit theorem.

math.PR↗

arXiv2022

Femtosecond Thermal and Nonthermal Hot Electron Tunneling inside a Photoexcited Tunnel Junction

Efficient operation of electronic nanodevices at ultrafast speeds requires understanding and control of the currents generated by femtosecond bursts of light. Ultrafast laser-induced currents in metallic nanojunctions can originate from photo-assisted hot electron tunneling or lightwave-induced tunneling. Both processes can drive localized photocurrents inside a scanning tunneling microscope (STM) on femto- to attosecond time scales, enabling ultrafast STM with atomic spatial resolution. Femtosecond laser excitation of a metallic nanojunction, however, also leads to the formation of a transient thermalized electron distribution, but the tunneling of thermalized hot electrons on time scales faster than electron-lattice equilibration is not well understood. Here, we investigate ultrafast electronic heating and transient thermionic tunneling inside a metallic photoexcited tunnel junction and its role in the generation of ultrafast photocurrents in STM. Phase-resolved sampling of broadband THz pulses via the THz-field-induced modulation of ultrafast photocurrents allows us to probe the electronic temperature evolution inside the STM tip, and to observe the competition between instantaneous and delayed tunneling due to nonthermal and thermal hot electron distributions in real time. Our results reveal the pronounced nonthermal character of photo-induced hot electron tunneling, and provide a detailed microscopic understanding of hot electron dynamics inside a laser-excited tunnel junction.

cond-mat.mes-hall↗

arXiv2022

Spectral dimension of simple random walk on a long-range percolation cluster

Consider the long-range percolation model on the integer lattice $\mathbb{Z}^d$ in which all nearest-neighbour edges are present and otherwise $x$ and $y$ are connected with probability $q_{x,y}:=1-\exp(-|x-y|^{-s})$, independently of the state of other edges. Throughout the regime where the model yields a locally-finite graph, (i.e.\ for $s>d$,) we determine the spectral dimension of the associated simple random walk, apart from at the exceptional value $d=1$, $s=2$, where the spectral dimension is discontinuous. Towards this end, we present various on-diagonal heat kernel bounds, a number of which are new. In particular, the lower bounds are derived through the application of a general technique that utilises the translation invariance of the model. We highlight that, applying this general technique, we are able to partially extend our main result beyond the nearest-neighbour setting, and establish lower heat kernel bounds over the range of parameters $s\in (d,2d)$. We further note that our approach is applicable to short-range models as well.

math.PR↗

arXiv2021

Heat kernel bounds for nonlocal operators with singular kernels

We prove sharp two-sided bounds of the fundamental solution for an integro-differential operator of order $α\in (0,2)$ that generates a $d$-dimensional Markov process. The corresponding Dirichlet form is comparable to that of $d$ independent copies of one-dimensional jump processes, i.e., the jumping measure is singular with respect to the $d$-dimensional Lebesgue measure.

math.AP↗

arXiv2021

Glauber dynamics for Ising models on random regular graphs: cut-off and metastability

Consider random $d$-regular graphs, i.e., random graphs such that there are exactly $d$ edges from each vertex for some $d\ge 3$. We study both the configuration model version of this graph, which has occasional multi-edges and self-loops, as well as the simple version of it, which is a $d$-regular graph chosen uniformly at random from the collection of all $d$-regular graphs. In this paper, we discuss mixing times of Glauber dynamics for the Ising model with an external magnetic field on a random $d$-regular graph, both in the quenched as well as the annealed settings. Let $β$ be the inverse temperature, $β_c$ be the critical temperature and $B$ be the external magnetic field. Concerning the annealed measure, we show that for $β> β_c$ there exists $\hat{B}_c(β)\in (0,\infty)$ such that the model is metastable (i.e., the mixing time is exponential in the graph size $n$) when $β> β_c$ and $0 \leq B < \hat{B}_c(β)$, whereas it exhibits the cut-off phenomenon at $c_\star n \log n$ with a window of order $n$ when $β< β_c$ or $β> β_c$ and $B>\hat{B}_c(β)$. Interestingly, $\hat{B}_c(β)$ coincides with the critical external field of the Ising model on the $d$-ary tree (namely, above which the model has a unique Gibbs measure). Concerning the quenched measure, we show that there exists $B_c(β)$ with $B_c(β) \leq \hat{B}_c(β)$ such that for $β> β_c$, the mixing time is at least exponential along some subsequence $(n_k)_{k\geq 1}$ when $0 \leq B < B_c(β)$, whereas it is less than or equal to $Cn\log n$ when $B>\hat{B}_c(β)$. The quenched results also hold for the model conditioned on simplicity, for the annealed results this is unclear.

math.PR↗

arXiv2021

Quenched Invariance Principle for a class of random conductance models with long-range jumps

We study random walks on $\mathbb Z^d$ (with $d\ge 2$) among stationary ergodic random conductances $\{C_{x,y}\colon x,y\in\mathbb Z^d\}$ that permit jumps of arbitrary length. Our focus is on the Quenched Invariance Principle (QIP) which we establish by a combination of corrector methods, functional inequalities and heat-kernel technology assuming that the $p$-th moment of $\sum_{x\in\mathbb Z^d}C_{0,x}|x|^2$ and $q$-th moment of $1/C_{0,x}$ for $x$ neighboring the origin are finite for some $p,q\ge1$ with $p^{-1}+q^{-1}<2/d$. In particular, a QIP thus holds for random walks on long-range percolation graphs with connectivity exponents larger than $2d$ in all $d\ge2$, provided all the nearest-neighbor edges are present. Although still limited by moment conditions, our method of proof is novel in that it avoids proving everywhere-sublinearity of the corrector. This is relevant because we show that, for long-range percolation with exponents between $d+2$ and $2d$, the corrector exists but fails to be sublinear everywhere. Similar examples are constructed also for nearest-neighbor, ergodic conductances in $d\ge3$ under the conditions complementary to those of the recent work of P. Bella and M. Schäffner. These examples elucidate the limitations of elliptic-regularity techniques that underlie much of the recent progress on these problems.

math.PR↗

arXiv2021

Quenched and averaged tails of the heat kernel of the two-dimensional uniform spanning tree

This article investigates the heat kernel of the two-dimensional uniform spanning tree. We improve previous work by demonstrating the occurrence of log-logarithmic fluctuations around the leading order polynomial behaviour for the on-diagonal part of the quenched heat kernel. In addition we give two-sided estimates for the averaged heat kernel, and we show that the exponents that appear in the off-diagonal parts of the quenched and averaged versions of the heat kernel differ. Finally, we derive various scaling limits for the heat kernel, the implications of which include enabling us to sharpen the known asymptotics regarding the on-diagonal part of the averaged heat kernel and the expected distance travelled by the associated simple random walk.

math.PR↗

arXiv2021

Heat kernels for reflected diffusions with jumps on inner uniform domains

In this paper, we study sharp two-sided heat kernel estimates for a large class of symmetric reflected diffusions with jumps on the closure of an inner uniform domain $D$ in a length metric space. The length metric is the intrinsic metric of a strongly local Dirichlet form. When $D$ is an inner uniform domain in the Euclidean space, a prototype for a special case of the processes under consideration are symmetric reflected diffusions with jumps on $D$, whose infinitesimal generators are non-local (pseudo-differential) operators $L$ on $D$ of the form $$ L u(x) =\frac12 \sum_{i, j=1}^d \frac{\partial}{\partial x_i} \left(a_{ij}(x) \frac{\partial u(x)}{\partial x_j}\right) + \lim_{\eps \downarrow 0} \int_{\{y\in D: \, ρ_D(y, x)>\eps\}} (u(y)-u(x)) J(x, y)\, dy $$ satisfying "Neumann boundary condition". Here, $ρ_D(x,y)$ is the length metric on $D$, $A(x)=(a_{ij}(x))_{1\leq i,j\leq d}$ is a measurable $d\times d$ matrix-valued function on $D$ that is uniformly elliptic and bounded, and $$ J(x,y):= \frac{1}{Φ(ρ_D(x,y))} \int_{[α_1, α_2]} \frac{c(α, x,y)} {ρ_D(x,y)^{d+α}} \,ν(dα) , $$ where $ν$ is a finite measure on $[α_1, α_2] \subset (0, 2)$, $Φ$ is an increasing function on $[ 0, \infty )$ with $c_1e^{c_2r^β} \le Φ(r) \le c_3 e^{c_4r^β}$ for some $β\in [0,\infty]$, and $c(α, x, y)$ is a jointly measurable function that is bounded between two positive constants and is symmetric in $(x, y)$.

math.PR↗

arXiv2020

Quenched invariance principle for long range random walks in balanced random environments

We establish via a probabilistic approach the quenched invariance principle for a class of long range random walks in independent (but not necessarily identically distributed) balanced random environments, with the transition probability from $x$ to $y$ on average being comparable to $|x-y|^{-(d+α)}$ with $α\in (0,2]$. We use the martingale property to estimate exit time from balls and establish tightness of the scaled processes, and apply the uniqueness of the martingale problem to identify the limiting process. When $α\in (0,1)$, our approach works even for non-balanced cases. When $α=2$, under a diffusive with the logarithmic perturbation scaling, we show that the limit of scaled processes is a Brownian motion.

math.PR↗

arXiv2020

Heat kernel upper bounds for symmetric Markov semigroups

It is well known that Nash-type inequalities for symmetric Dirichlet forms are equivalent to on-diagonal heat kernel upper bounds for the associated symmetric Markov semigroups. In this paper, we show that both imply (and hence are equivalent to) off-diagonal heat kernel upper bounds under some mild assumptions. Our approach is based on a new generalized Davies' method. Our results extend that of \cite{CKS} for Nash-type inequalities with power order considerably and also extend that of \cite{Gri} for second order differential operators on a complete non-compact manifold.

math.PR↗

arXiv2020

Periodic homogenization of non-symmetric Lévy-type processes

In this paper, we study homogenization problem for strong Markov processes on $\R^d$ having infinitesimal generators $$ \sL f(x)=\int_{\R^d}\left(f(x+z)-f(x)-\langle \nabla f(x), z\rangle \I_{\{|z|\le 1\}} \right) k(x,z)\, Π(dz) +\langle b(x), \nabla f(x) \rangle, \quad f\in C^2_b (\R^d) $$ in periodic media, where $Π$ is a non-negative measure on $\R^d$ that does not charge the origin $0$, satisfies $\int_{\R^d} (1 \wedge |z|^2)\, Π(dz)<\infty$, and can be singular with respect to the Lebesgue measure on $\R^d$. Under a proper scaling, we show the scaled processes converge weakly to Lévy processes on $\R^d$. The results are a counterpart of the celebrated work \cite{BLP,Bh} in the jump-diffusion setting. In particular, we completely characterize the homogenized limiting processes when $b(x) $ is a bounded continuous multivariate 1-periodic $\R^d$-valued function, $k(x,z)$ is a non-negative bounded continuous function that is multivariate 1-periodic in both $x$ and $z$ variables, and, in spherical coordinate $z=(r, θ) \in \R_+\times \bS^{d-1}$, $$ \I_{\{|z|>1\}}\,Π(dz) = \I_{\{ r>1\}} \varrho_0(dθ) \, \frac{ dr }{r^{1+α}} $$ with $α\in (0,\infty)$ and $\varrho_0$ being any finite measure on the unit sphere $\bS^{d-1}$ in $\R^d$. Different phenomena occur depending on the values of $α$; there are five cases: $α\in (0, 1)$, $α=1$, $α\in (1, 2)$, $α=2$ and $α\in (2, \infty)$.

math.PR↗

arXiv2020

Stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms

In this paper, we establish stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms on metric measure spaces under general volume doubling condition. We obtain their stable equivalent characterizations in terms of the jumping kernels, variants of cutoff Sobolev inequalities, and Poincaré inequalities. In particular, we establish the connection between parabolic Harnack inequalities and two-sided heat kernel estimates, as well as with the Hölder regularity of parabolic functions for symmetric non-local Dirichlet forms.

math.PR↗

arXiv2020

Random Conductance Models with Stable-like Jumps: Quenched Invariance Principle

We study the quenched invariance principle for random conductance models with long range jumps on $\Z^d$, where the transition probability from $x$ to $y$ is, on average, comparable to $|x-y|^{-(d+α)}$ with $α\in (0,2)$ but is allowed to be degenerate. Under some moment conditions on the conductance, we prove that the scaling limit of the Markov process is a symmetric $α$-stable Lévy process on $\R^d$.The well-known corrector method in homogenization theory does not seem to work in this setting. Instead, we utilize probabilistic potential theory for the corresponding jump processes. Two essential ingredients of our proof are the tightness estimate and the Hölder regularity of caloric functions for non-elliptic $α$-stable-like processes on graphs. Our method is robust enough to apply not only for $\Z^d$ but also for more general graphs whose scaling limits are nice metric measure spaces.

math.PR↗

arXiv2020

Homogenization of symmetric stable-like processes in stationary ergodic medium

This paper studies homogenization of symmetric non-local Dirichlet forms with $α$-stable-like jumping kernels in one-parameter stationary ergodic environment. Under suitable conditions, we establish homogenization results and identify the limiting effective Dirichlet forms explicitly. The coefficients of the jumping kernels of Dirichlet forms and symmetrizing measures are allowed to be degenerate and unbounded; and the coefficients in the effective Dirichlet forms can be degenerate.

math.PR↗

arXiv2019

Heat kernel estimates for general symmetric pure jump Dirichlet forms

In this paper, we consider the following symmetric non-local Dirichlet forms of pure jump type on metric measure space $(M,d,μ)$: $$\mathcal{E}(f,g)=\int_{M\times M} (f(x)-f(y))(g(x)-g(y))\,J(dx,dy),$$ where $J(dx,dy)$ is a symmetric Radon measure on $M\times M\setminus {\rm diag}$ that may have different scalings for small jumps and large jumps. Under general volume doubling condition on $(M,d,μ)$ and some mild quantitative assumptions on $J(dx, dy)$ that are allowed to have light tails of polynomial decay at infinity, we establish stability results for two-sided heat kernel estimates as well as heat kernel upper bound estimates in terms of jumping kernel bounds, the cut-off Sobolev inequalities, and the Faber-Krahn inequalities (resp.\ the Poincaré inequalities). We also give stable characterizations of the corresponding parabolic Harnack inequalities.

math.PR↗

arXiv2019

Heat kernel estimates and parabolic Harnack inequalities for symmetric Dirichlet forms

In this paper, we consider the following symmetric Dirichlet forms on a metric measure space $(M,d,μ)$: $$\mathcal{E}(f,g) = \mathcal{E}(^{(c)}(f,g)+\int_{M\times M} (f(x)-f(y))(g(x)-g(y))\,J(dx,dy),$$ where $\mathcal{E}(^{(c)}$ is a strongly local symmetric bilinear form and $J(dx,dy)$ is a symmetric Random measure on $M\times M$. Under general volume doubling condition on $(M,d,μ)$ and some mild assumptions on scaling functions, we establish stability results for upper bounds of heat kernel (resp.\ two-sided heat kernel estimates) in terms of the jumping kernels, the cut-off Sobolev inequalities, and the Faber-Krahn inequalities (resp.\ the Poincaré inequalities). We also obtain characterizations of parabolic Harnack inequalities. Our results apply to symmetric diffusions with jumps even when the underlying spaces have walk dimensions larger than $2$.

math.PR↗

arXiv2019

Elucidating the NuclearQuantum Dynamics of Intramolecular Double Hydrogen Transfer in Porphycene

We address the double hydrogen transfer (DHT) dynamics of the porphycene molecule: A complex paradigmatic system where the making and breaking of H-bonds in a highly anharmonic potential energy surface requires a quantum mechanical treatment not only of the electrons, but also of the nuclei. We combine density-functional theory calculations, employing hybrid functionals and van der Waals corrections, with recently proposed and optimized path-integral ring-polymer methods for the approximation of quantum vibrational spectra and reaction rates. Our full-dimensional ring-polymer instanton simulations show that below 100 K the concerted DHT tunneling pathway dominates, but between 100 K and 300 K there is a competition between concerted and stepwise pathways when nuclear quantum effects are included. We obtain ground-state reaction rates of $2.19 \times 10^{11} \mathrm{s}^{-1}$ at 150 K and $0.63 \times 10^{11} \mathrm{s}^{-1}$ at 100 K, in good agreement with experiment. We also reproduce the puzzling N-H stretching band of porphycene with very good accuracy from thermostatted ring-polymer molecular dynamics simulations. The position and lineshape of this peak, centered at around 2600 cm$^{-1}$ and spanning 750 cm$^{-1}$, stems from a combination of very strong H-bonds, the coupling to low-frequency modes, and the access to $cis$-like isomeric conformations, which cannot be appropriately captured with classical-nuclei dynamics. These results verify the appropriateness of our general theoretical approach and provide a framework for a deeper physical understanding of hydrogen transfer dynamics in complex systems.

physics.chem-ph↗

arXiv2018

Time Fractional Poisson Equations: Representations and Estimates

In this paper, we study existence and uniqueness of strong as well as weak solutions for general time fractional Poisson equations. We show that there is an integral representation of the solutions of time fractional Poisson equations with zero initial values in terms of semigroup for the infinitesimal spatial generator ${\cal L}$ and the corresponding subordinator associated with the time fractional derivative. This integral representation has an integral kernel $q(t, x, y)$, which we call the fundamental solution for the time fractional Poisson equation, if the semigrou for ${\cal L}$ has an integral kernel. We further show that $q(t, x, y)$ can be expressed as a time fractional derivative of the fundamental solution for the homogenous time fractional equation under the assumption that the associated subordinator admits a conjugate subordinator. Moreover, when the Laplace exponent of the associated subordinator satisfies the weak scaling property and its distribution is self-decomposable, we establish two-sided estimates for the fundamental solution $q(t,x, y)$ through explicit estimates of transition density functions of subordinators.

math.AP↗