اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدAsymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation
180
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
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 $|x|^{-d-alpha}$ with $alpha>0$. For random walk in any dimension $d$ and for self-avoiding walk and critical/subcritical oriented percolation above the common upper-critical dimension $d_{mathrm{c}}equiv2(alphawedge2)$, 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. Poincar{e} 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].
قيم البحث
اقرأ أيضاً
We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if $d>2(alphawedge2)$ for self-avoiding walk and the Ising model, and $d>3(alphaw
edge2)$ for percolation, where $d$ denotes the dimension and $alpha$ the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. (2007)
We consider a long-range version of self-avoiding walk in dimension $d > 2(alpha wedge 2)$, where $d$ denotes dimension and $alpha$ the power-law decay exponent of the coupling function. Under appropriate scaling we prove convergence to Brownian moti
on for $alpha ge 2$, and to $alpha$-stable Levy motion for $alpha < 2$. This complements results by Slade (1988), who proves convergence to Brownian motion for nearest-neighbor self-avoiding walk in high dimension.
We prove that the Fourier transform of the properly-scaled normalized two-point function for sufficiently spread-out long-range oriented percolation with index alpha>0 converges to e^{-C|k|^{alphawedge2}} for some Cin(0,infty) above the upper-critica
l dimension 2(alphawedge2). This answers the open question remained in the previous paper [arXiv:math/0703455]. Moreover, we show that the constant C exhibits crossover at alpha=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.
The aim of this short article is to convey the basic idea of the original paper [3], without going into too much detail, about how to derive sharp asymptotics of the gyration radius for random walk, self-avoiding walk and oriented percolation above the model-dependent upper critical dimension.
We consider instances of long-range percolation on $mathbb Z^d$ and $mathbb R^d$, where points at distance $r$ get connected by an edge with probability proportional to $r^{-s}$, for $sin (d,2d)$, and study the asymptotic of the graph-theoretical (a.
k.a. chemical) distance $D(x,y)$ between $x$ and $y$ in the limit as $|x-y|toinfty$. For the model on $mathbb Z^d$ we show that, in probability as $|x|toinfty$, the distance $D(0,x)$ is squeezed between two positive multiples of $(log r)^Delta$, where $Delta:=1/log_2(1/gamma)$ for $gamma:=s/(2d)$. For the model on $mathbb R^d$ we show that $D(0,xr)$ is, in probability as $rtoinfty$ for any nonzero $xinmathbb R^d$, asymptotic to $phi(r)(log r)^Delta$ for $phi$ a positive, continuous (deterministic) function obeying $phi(r^gamma)=phi(r)$ for all $r>1$. The proof of the asymptotic scaling is based on a subadditive argument along a continuum of doubly-exponential sequences of scales. The results strengthen considerably the conclusions obtained earlier by the first author. Still, significant open questions remain.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
