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Register a new userOn the exponential growth rates of lattice animals and interfaces II: new asymptotic bounds
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Added by
Christoforos Panagiotis
Publication date
2021
fields
Physics
and research's language is
English
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We introduce a method for translating any upper bound on the percolation threshold of a lattice $G$ into a lower bound on the exponential growth rate $a(G)$ of lattice animals and vice-versa. We exploit this in both directions. We improve on the best known asymptotic lower and upper bounds on $a(mathbb{Z}^d)$ as $dto infty$. We use percolation as a tool to obtain the latter, and conversely we use the former to obtain lower bounds on $p_c(mathbb{Z}^d)$. We obtain the rigorous lower bound $dot{p}_c(mathbb{Z}^3)>0.2522$ for 3-dimensional site percolation.
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We introduce a formula for translating any upper bound on the percolation threshold of a lattice g into a lower bound on the exponential growth rate of lattice animals $a(G)$ and vice-versa. We exploit this to improve on the best known asymptotic bounds on $a(mathbb{Z}^d)$ as $dto infty$. Our formula remains valid if instead of lattice animals we enumerate certain sub-species called interfaces. Enumerating interfaces leads to functional duality formulas that are tightly connected to percolation and are not valid for lattice animals, as well as to strict inequalities for the percolation threshold. Incidentally, we prove that the rate of the exponential decay of the cluster size distribution of Bernoulli percolation is a continuous function of $pin (0,1)$.
A well-known question in the planar first-passage percolation model concerns the convergence of the empirical distribution along geodesics. We demonstrate this convergence for an explicit model, directed last-passage percolation on $mathbb{Z}^2$ with i.i.d. exponential weights, and provide explicit formulae for the limiting distributions, which depend on the asymptotic direction. For example, for geodesics in the direction of the diagonal, the limiting weight distribution has density $(1/4+x/2+x^2/8)e^{-x}$, and so is a mixture of Gamma($1,1$), Gamma($2,1$) and Gamma($3,1$) distributions with weights $1/4$, $1/2$, and $1/4$ respectively. More generally, we study the local environment as seen from vertices along the geodesics (including information about the shape of the path and about the weights on and off the path in a local neighborhood). We consider finite geodesics from $(0,0)$ to $nboldsymbol{rho}$ for some vector $boldsymbol{rho}$ in the first quadrant, in the limit as $ntoinfty$, as well as the semi-infinite geodesic in direction $boldsymbol{rho}$. We show almost sure convergence of the empirical distributions along the geodesic, as well as convergence of the distribution around a typical point, and we give an explicit description of the limiting distribution. We make extensive use of a correspondence with TASEP as seen from a single second-class particle for which we prove new results concerning ergodicity and convergence to equilibrium. Our analysis relies on geometric arguments involving estimates for the last-passage time, available from the integrable probability literature.
We study the stochastic growth process in discrete time $x_{i+1} = (1 + mu_i) x_i$ with growth rate $mu_i = rho e^{Z_i - frac12 var(Z_i)}$ proportional to the exponential of an Ornstein-Uhlenbeck (O-U) process $dZ_t = - gamma Z_t dt + sigma dW_t$ sampled on a grid of uniformly spaced times ${t_i}_{i=0}^n$ with time step $tau$. Using large deviation theory methods we compute the asymptotic growth rate (Lyapunov exponent) $lambda = lim_{nto infty} frac{1}{n} log mathbb{E}[x_n]$. We show that this limit exists, under appropriate scaling of the O-U parameters, and can be expressed as the solution of a variational problem. The asymptotic growth rate is related to the thermodynamical pressure of a one-dimensional lattice gas with attractive exponential potentials. For $Z_t$ a stationary O-U process the lattice gas coincides with a system considered previously by Kac and Helfand. We derive upper and lower bounds on $lambda$. In the large mean-reversion limit $gamma n tau gg 1$ the two bounds converge and the growth rate is given by a lattice version of the van der Waals equation of state. The predictions are tested against numerical simulations of the stochastic growth model.
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.
We propose a twisted Szegedy walk for estimating the limit behavior of a discrete-time quantum walk on a crystal lattice, an infinite abelian covering graph, whose notion was introduced by [14]. First, we show that the spectrum of the twisted Szegedy walk on the quotient graph can be expressed by mapping the spectrum of a twisted random walk onto the unit circle. Secondly, we show that the spatial Fourier transform of the twisted Szegedy walk on a finite graph with appropriate parameters becomes the Grover walk on its infinite abelian covering graph. Finally, as an application, we show that if the Betti number of the quotient graph is strictly greater than one, then localization is ensured with some appropriated initial state. We also compute the limit density function for the Grover walk on $mathbb{Z}^d$ with flip flop shift, which implies the coexistence of linear spreading and localization. We partially obtain the abstractive shape of the limit density function: the support is within the $d$-dimensional sphere of radius $1/sqrt{d}$, and $2^d$ singular points reside on the spheres surface.
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