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Register a new userAnalyticity of the percolation density $theta$ in all dimensions
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Added by
Christoforos Panagiotis
Publication date
2020
fields
Physics
and research's language is
English
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We prove that for Bernoulli bond percolation on $mathbb{Z}^d$, $dgeq 2$ the percolation density is an analytic function of the parameter in the supercritical interval $(p_c,1]$. This answers a question of Kesten from 1981.
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We prove that for Bernoulli percolation on $mathbb{Z}^d$, $dgeq 2$, the percolation density is an analytic function of the parameter in the supercritical interval. For this we introduce some techniques that have further implications. In particular, we prove that the susceptibility is analytic in the subcritical interval for all transitive short- or long-range models, and that $p_c^{bond} <1/2$ for certain families of triangulations for which Benjamini & Schramm conjectured that $p_c^{site} leq 1/2$.
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 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.
The arboreal gas is the probability measure on (unrooted spanning) forests of a graph in which each forest is weighted by a factor $beta>0$ per edge. It arises as the $qto 0$ limit with $p=beta q$ of the $q$-state random cluster model. We prove that in dimensions $dgeq 3$ the arboreal gas undergoes a percolation phase transition. This contrasts with the case of $d=2$ where all trees are finite for all $beta>0$. The starting point for our analysis is an exact relationship between the arboreal gas and a fermionic non-linear sigma model with target space $mathbb{H}^{0|2}$. This latter model can be thought of as the $0$-state Potts model, with the arboreal gas being its random cluster representation. Unlike the $q>0$ Potts models, the $mathbb{H}^{0|2}$ model has continuous symmetries. By combining a renormalisation group analysis with Ward identities we prove that this symmetry is spontaneously broken at low temperatures. In terms of the arboreal gas, this symmetry breaking translates into the existence of infinite trees in the thermodynamic limit. Our analysis also establishes massless free field correlations at low temperatures and the existence of a macroscopic tree on finite tori.
Consider nearest-neighbor oriented percolation in $d+1$ space-time dimensions. Let $rho,eta, u$ be the critical exponents for the survival probability up to time $t$, the expected number of vertices at time $t$ connected from the space-time origin, and the gyration radius of those vertices, respectively. We prove that the hyperscaling inequality $d ugeeta+2rho$, which holds for all $dge1$ and is a strict inequality above the upper-critical dimension 4, becomes an equality for $d=1$, i.e., $ u=eta+2rho$, provided existence of at least two among $rho,eta, u$. The key to the proof is the recent result on the critical box-crossing property by Duminil-Copin, Tassion and Teixeira (2017).
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