arXiv subjects

Lung-Chi Chen

Publications and source records attributed to Lung-Chi Chen.

16 recordsLinked to original sources

arXiv2022

Mean-field behavior of nearest-neighbor oriented percolation on the BCC lattice above $8+1$ dimensions

In this paper, we consider nearest-neighbor oriented percolation with independent Bernoulli bond-occupation probability on the $d$-dimensional body-centered cubic (BCC) lattice $\mathbb{L}^d$ and the set of non-negative integers $\mathbb{Z}_+$. Thanks to the orderly structure of the BCC lattice, we prove that the infrared bound holds on $\mathbb{L}^d\times\mathbb{Z}_+$ in all dimensions $d\geq 9$. As opposed to ordinary percolation, we have to deal with complex numbers due to asymmetry induced by time-orientation, which makes it hard to bound the bootstrap functions in the lace-expansion analysis. By investigating the Fourier-Laplace transform of the random-walk Green function and the two-point function, we derive the key properties to obtain the upper bounds and resolve a problematic issue in Nguyen and Yang's bound. The issue is caused by the fact that the Fourier transform of the random-walk transition probability can take the value $-1$.

math-ph↗

arXiv2019

Critical two-point function for long-range models with power-law couplings: The marginal case for $d\ge d_c$

Consider the long-range models on $\mathbb{Z}^d$ of random walk, self-avoiding walk, percolation and the Ising model, whose translation-invariant 1-step distribution/coupling coefficient decays as $|x|^{-d-α}$ for some $α>0$. In the previous work (Ann. Probab., 43, 639--681, 2015), we have shown in a unified fashion for all $α\ne2$ that, assuming a bound on the "derivative" of the $n$-step distribution (the compound-zeta distribution satisfies this assumed bound), the critical two-point function $G_{p_c}(x)$ decays as $|x|^{α\wedge2-d}$ above the upper-critical dimension $d_c\equiv(α\wedge2)m$, where $m=2$ for self-avoiding walk and the Ising model and $m=3$ for percolation. In this paper, we show in a much simpler way, without assuming a bound on the derivative of the $n$-step distribution, that $G_{p_c}(x)$ for the marginal case $α=2$ decays as $|x|^{2-d}/\log|x|$ whenever $d\ge d_c$ (with a large spread-out parameter $L$). This solves the conjecture in the previous work, extended all the way down to $d=d_c$, and confirms a part of predictions in physics (Brezin, Parisi, Ricci-Tersenghi, J. Stat. Phys., 157, 855--868, 2014). The proof is based on the lace expansion and new convolution bounds on power functions with log corrections.

math-ph↗

arXiv2015

Critical two-point functions for long-range statistical-mechanical models in high dimensions

We consider long-range self-avoiding walk, percolation and the Ising model on $\mathbb{Z}^d$ that are defined by power-law decaying pair potentials of the form $D(x)\asymp|x|^{-d-α}$ with $α>0$. The upper-critical dimension $d_{\mathrm{c}}$ is $2(α\wedge2)$ for self-avoiding walk and the Ising model, and $3(α\wedge2)$ for percolation. Let $α\ne2$ and assume certain heat-kernel bounds on the $n$-step distribution of the underlying random walk. We prove that, for $d>d_{\mathrm{c}}$ (and the spread-out parameter sufficiently large), the critical two-point function $G_{p_{\mathrm{c}}}(x)$ for each model is asymptotically $C|x|^{α\wedge2-d}$, where the constant $C\in(0,\infty)$ is expressed in terms of the model-dependent lace-expansion coefficients and exhibits crossover between $α<2$ and $α>2$. We also provide a class of random walks that satisfy those heat-kernel bounds.

math-ph↗

arXiv2012

A Monotonicity Result for the Range of a Perturbed Random Walk

We consider a discrete time simple symmetric random walk on Z^d, d>=1, where the path of the walk is perturbed by inserting deterministic jumps. We show that for any time n and any deterministic jumps that we insert, the expected number of sites visited by the perturbed random walk up to time n is always larger than or equal to that for the unperturbed walk. This intriguing problem arises from the study of a particle among a Poisson system of moving traps with sub-diffusive trap motion. In particular, our result implies a variant of the Pascal principle, which asserts that among all deterministic trajectories the particle can follow, the constant trajectory maximizes the particle's survival probability up to any time t>0.

math.PR↗

arXiv2012

Ice model and eight-vertex model on the two-dimensional Sierpinski gasket

We present the numbers of ice model and eight-vertex model configurations (with Boltzmann factors equal to one), I(n) and E(n) respectively, on the two-dimensional Sierpinski gasket SG(n) at stage $n$. For the eight-vertex model, the number of configurations is $E(n)=2^{3(3^n+1)/2}$ and the entropy per site, defined as $\lim_{v \to \infty} \ln E(n)/v$ where $v$ is the number of vertices on SG(n), is exactly equal to $\ln 2$. For the ice model, the upper and lower bounds for the entropy per site $\lim_{v \to \infty} \ln I(n)/v$ are derived in terms of the results at a certain stage. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical value of the entropy can be evaluated with more than a hundred significant figures accurate. The corresponding result of ice model on the generalized two-dimensional Sierpinski gasket SG_b(n) with $b=3$ is also obtained. For the generalized vertex model on SG_3(n), the number of configurations is $2^{(8 \times 6^n +7)/5}$ and the entropy per site is equal to $\frac87 \ln 2$. The general upper and lower bounds for the entropy per site for arbitrary $b$ are conjectured.

cond-mat.stat-mech↗

arXiv2011

Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation

We consider random walk and self-avoiding walk whose 1-step distribution is given by $D$, and oriented percolation whose bond-occupation probability is proportional to $D$. Suppose that $D(x)$ decays as $|x|^{-d-α}$ with $α>0$. For random walk in any dimension $d$ and for self-avoiding walk and critical/subcritical oriented percolation above the common upper-critical dimension $d_{\mathrm{c}}\equiv2(α\wedge2)$, we prove large-$t$ asymptotics of the gyration radius, which is the average end-to-end distance of random walk/self-avoiding walk of length $t$ or the average spatial size of an oriented percolation cluster at time $t$. This proves the conjecture for long-range self-avoiding walk in [Ann. Inst. H. Poincaré Probab. Statist. (2010), to appear] and for long-range oriented percolation in [Probab. Theory Related Fields 142 (2008) 151--188] and [Probab. Theory Related Fields 145 (2009) 435--458].

math.PR↗

arXiv2011

Asymptotic enumeration of independent sets on the Sierpinski gasket

The number of independent sets is equivalent to the partition function of the hard-core lattice gas model with nearest-neighbor exclusion and unit activity. We study the number of independent sets $m_{d,b}(n)$ on the generalized Sierpinski gasket $SG_{d,b}(n)$ at stage $n$ with dimension $d$ equal to two, three and four for $b=2$, and layer $b$ equal to three for $d=2$. The upper and lower bounds for the asymptotic growth constant, defined as $z_{SG_{d,b}}=\lim_{v \to \infty} \ln m_{d,b}(n)/v$ where $v$ is the number of vertices, on these Sierpinski gaskets are derived in terms of the results at a certain stage. The numerical values of these $z_{SG_{d,b}}$ are evaluated with more than a hundred significant figures accurate. We also conjecture the upper and lower bounds for the asymptotic growth constant $z_{SG_{d,2}}$ with general $d$.

cond-mat.stat-mech↗

arXiv2009

Hamiltonian paths on the Sierpinski gasket

We derive exactly the number of Hamiltonian paths H(n) on the two dimensional Sierpinski gasket SG(n) at stage $n$, whose asymptotic behavior is given by $\frac{\sqrt{3}(2\sqrt{3})^{3^{n-1}}}{3} \times (\frac{5^2 \times 7^2 \times 17^2}{2^{12} \times 3^5 \times 13})(16)^n$. We also obtain the number of Hamiltonian paths with one end at a certain outmost vertex of SG(n), with asymptotic behavior $\frac {\sqrt{3}(2\sqrt{3})^{3^{n-1}}}{3} \times (\frac {7 \times 17}{2^4 \times 3^3})4^n$. The distribution of Hamiltonian paths on SG(n) with one end at a certain outmost vertex and the other end at an arbitrary vertex of SG(n) is investigated. We rigorously prove that the exponent for the mean $\ell$ displacement between the two end vertices of such Hamiltonian paths on SG(n) is $\ell \log 2 / \log 3$ for $\ell>0$.

cond-mat.stat-mech↗

arXiv2008

Critical behavior and the limit distribution for long-range oriented percolation. II: Spatial correlation

We prove that the Fourier transform of the properly-scaled normalized two-point function for sufficiently spread-out long-range oriented percolation with index α>0 converges to e^{-C|k|^{α\wedge2}} for some C\in(0,\infty) above the upper-critical dimension 2(α\wedge2). This answers the open question remained in the previous paper [arXiv:math/0703455]. Moreover, we show that the constant C exhibits crossover at α=2, which is a result of interactions among occupied paths. The proof is based on a new method of estimating fractional moments for the spatial variable of the lace-expansion coefficients.

math.PR↗

arXiv2008

Number of connected spanning subgraphs on the Sierpinski gasket

We study the number of connected spanning subgraphs $f_{d,b}(n)$ on the generalized Sierpinski gasket $SG_{d,b}(n)$ at stage $n$ with dimension $d$ equal to two, three and four for $b=2$, and layer $b$ equal to three and four for $d=2$. The upper and lower bounds for the asymptotic growth constant, defined as $z_{SG_{d,b}}=\lim_{v \to \infty} \ln f_{d,b}(n)/v$ where $v$ is the number of vertices, on $SG_{2,b}(n)$ with $b=2,3,4$ are derived in terms of the results at a certain stage. The numerical values of $z_{SG_{d,b}}$ are obtained.

math-ph↗

arXiv2008

Structure of spanning trees on the two-dimensional Sierpinski gasket

Consider spanning trees on the two-dimensional Sierpinski gasket SG(n) where stage $n$ is a non-negative integer. For any given vertex $x$ of SG(n), we derive rigorously the probability distribution of the degree $j \in \{1,2,3,4\}$ at the vertex and its value in the infinite $n$ limit. Adding up such probabilities of all the vertices divided by the number of vertices, we obtain the average probability distribution of the degree $j$. The corresponding limiting distribution $ϕ_j$ gives the average probability that a vertex is connected by 1, 2, 3 or 4 bond(s) among all the spanning tree configurations. They are rational numbers given as $ϕ_1=10957/40464$, $ϕ_2=6626035/13636368$, $ϕ_3=2943139/13636368$, $ϕ_4=124895/4545456$.

math-ph↗

arXiv2007

Dimer coverings on the Sierpinski gasket with possible vacancies on the outmost vertices

We present the number of dimers $N_d(n)$ on the Sierpinski gasket $SG_d(n)$ at stage $n$ with dimension $d$ equal to two, three, four or five, where one of the outmost vertices is not covered when the number of vertices $v(n)$ is an odd number. The entropy of absorption of diatomic molecules per site, defined as $S_{SG_d}=\lim_{n \to \infty} \ln N_d(n)/v(n)$, is calculated to be $\ln(2)/3$ exactly for $SG_2(n)$. The numbers of dimers on the generalized Sierpinski gasket $SG_{d,b}(n)$ with $d=2$ and $b=3,4,5$ are also obtained exactly. Their entropies are equal to $\ln(6)/7$, $\ln(28)/12$, $\ln(200)/18$, respectively. The upper and lower bounds for the entropy are derived in terms of the results at a certain stage for $SG_d(n)$ with $d=3,4,5$. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical value of $S_{SG_d}$ with $d=3,4,5$ can be evaluated with more than a hundred significant figures accurate.

cond-mat.stat-mech↗

arXiv2007

Critical behavior and the limit distribution for long-range oriented percolation. I

We consider oriented percolation on Z^d times Z_+ whose bond-occupation probability is pD(...), where p is the percolation parameter and D is a probability distribution on Z^d. Suppose that D(x) decays as |x|^{-d-α} for some α>0. We prove that the two-point function obeys an infrared bound which implies that various critical exponents take on their respective mean-field values above the upper-critical dimension 2\min{α,2}. We also show that, for every k, the Fourier transform of the normalized two-point function at time n, with a proper spatial scaling, has a convergent subsequence to exp(-c|k|^{\min{α,2}}) for some c>0.

math.PR↗

arXiv2007

Dimer-monomer model on the Sierpinski gasket

We present the numbers of dimer-monomers on the Sierpinski gasket $SG_d(n)$ at stage $n$ with dimension $d$ equal to two, three and four, and determine the asymptotic behaviors. The corresponding results on the generalized Sierpinski gasket $SG_{d,b}(n)$ with $d=2$ and $b=3,4$ are obtained.

cond-mat.stat-mech↗

arXiv2006

Spanning forests on the Sierpinski gasket

We present the numbers of spanning forests on the Sierpinski gasket $SG_d(n)$ at stage $n$ with dimension $d$ equal to two, three and four, and determine the asymptotic behaviors. The corresponding results on the generalized Sierpinski gasket $SG_{d,b}(n)$ with $d=2$ and $b=3,4$ are obtained. We also derive the upper bounds of the asymptotic growth constants for both $SG_d$ and $SG_{2,b}$.

math-ph↗

arXiv2006

Spanning trees on the Sierpinski gasket

We obtain the numbers of spanning trees on the Sierpinski gasket $SG_d(n)$ with dimension $d$ equal to two, three and four. The general expression for the number of spanning trees on $SG_d(n)$ with arbitrary $d$ is conjectured. The numbers of spanning trees on the generalized Sierpinski gasket $SG_{d,b}(n)$ with $d=2$ and $b=3,4$ are also obtained.

cond-mat.stat-mech↗