arXiv subjects

Christoforos Panagiotis

Publications and source records attributed to Christoforos Panagiotis.

At least 19 recordsLinked to original sources

arXiv2026

Sharpness and critical scaling of parking

In the parking model, each site of the $d$-dimensional lattice independently starts with one car with probability $p$ or one parking spot with probability $1-p$. Cars move according to independent discrete-time simple random walks and park at the first spot they find free. We prove that in the critical regime $p=1/2$, the expected number of visits to a site in $n$ rounds is of order $n^{(4-d)/4}$ for $d\leq3$ and $\log n$ for $d\geq4$. We also prove that in the subcritical regime $p\in(0,1/2)$, the parking-time tail is bounded above and below by stretched exponentials with exponent $d/(d+2)$. As $p\uparrow1/2$, we also determine the divergence of the expected total number of visits to a site: its order is $(1-2p)^{-3}$, $(1-2p)^{-1}$ and $(1-2p)^{-1/3}$ in dimensions one, two and three, respectively, and $\log(1/(1-2p))$ in dimensions four and higher. Our proof uses a representation of the parking process as the divisible sandpile of Levine and Peres plus a martingale-type term. These results answer questions posed by Damron, Gravner, Junge, Lyu and Sivakoff (2019).

math.PR↗

arXiv2026

Quantitative explosion and percolation of the divisible sandpile

The divisible sandpile on $\mathbb{Z}^d$ starts from i.i.d. masses at each site, and, in each discrete time step, a site with mass above one keeps one unit and sends the excess equally to its neighbors. Levine, Murugan, Peres and Ugurcan (2016) showed that at mean one this process explodes, with every site emitting infinite mass. We show that the mass emitted from a site by time $t$ is of order $t^{(4-d)/4}$ for $d\leq3$, of order $\log t$ for $d=4$, and a tail-dependent, divergent rate for $d\geq5$. We further show that the mass emitted, after diffusive rescaling, converges to a Brownian optimal-stopping value for $d\leq3$ and to tail-dependent, weighted membrane fields for $d\geq5$, while at the critical dimension $d=4$, after superdiffusive rescaling, it converges to the membrane model. Using these estimates, we prove that, for every $d\geq2$, the set of sites that topple contains an infinite component at some mean below one, hence it has a non-trivial percolation phase transition. This answers a variant of a question of Fey, Meester and Redig (2009). The proof adapts ideas from the theory of level-set percolation of strongly correlated Gaussian fields.

math.PR↗

arXiv2026

Supercritical sharpness for the random cluster representation of real-valued spin models

We study the supercritical regime $β>β_c$ of a large family of real-valued spin models on $\mathbb Z^d$, including the Blume-Capel and $P(φ)$ models. We consider their random cluster representations and prove that they are well-behaved, in the sense that local uniqueness of macroscopic clusters occurs with high probability, uniformly in the boundary conditions. This implies, among other things, a surface-order exponential bound for the (lower) large deviations of the empirical magnetisation. These results were previously known only in the cases of the Ising and $φ^4$ models.

math.PR↗

arXiv2026

Regularity of Gibbs measures for unbounded spin systems on general graphs

We consider a general class of spin systems with potentially unbounded real-valued spins, defined via a single-site potential with super-Gaussian tails on general graphs, allowing for both short- and long-range interactions. This class includes all $P(φ)$ models, in particular the well-studied $φ^4$ model. We construct an infinite-volume extremal measure called the plus measure as the limit of finite-volume Gibbs measures with weakly growing boundary conditions and show that it is regular, in the sense that it admits a bounded Radon-Nikodym derivative with respect to a product measure of single-site distributions with super-Gaussian tails. Moreover, we provide an alternative construction of the plus measure as the limit of finite-volume Gibbs measures that are regular up to the boundary. As a key intermediate step, we establish regularity and tightness of finite-volume Gibbs measures for a large class of growing boundary conditions $ξ$. Our regularity estimates are encoded in terms of a function $A(ξ)$, which provides precise control on the change of measure induced by boundary perturbations, and can thus be viewed as an analogue of the Cameron-Martin theorem for non-Gaussian fields. In the nearest-neighbour case, this class includes boundary conditions that grow at most double-exponentially in the distance to the boundary when the single-site measure has tails of the form $e^{-a|u|^n}$ for some $n>2$.Our results apply to arbitrary graphs and improve upon earlier results of Lebowitz and Presutti, and Ruelle, which apply in the context of $\mathbb{Z}^d$ and allow only logarithmically growing boundary conditions, as well as subsequent extensions to vertex-transitive graphs of polynomial growth.

math.PR↗

arXiv2026

Subcritical sharpness for real-valued spin models

In this paper, we consider a large family of real-valued spin models on general transitive graphs. We show that, in the subcritical regime $β<β_c$, the correlations of the model decay exponentially fast. To prove this result, we consider the random cluster (a.k.a. FK percolation) representation of the model and obtain an inequality that generalises the OSSS inequality to monotonic measures with random connection probabilities, thus extending the inequality of Duminil-Copin, Raoufi, and Tassion. Our results apply in particular to the Blume--Capel model and general $P(φ)$ models, going beyond the cases of the Ising and $φ^4$ models treated by Aizenman, Barsky, and Fernández.

math.PR↗

arXiv2026

Uniqueness of the infinite cluster for monotone percolation models without insertion tolerance

We consider a broad class of dependent site-percolation models on $\mathbb{Z}^d$ obtained by applying a monotone automaton to a random initial particle configuration drawn from a stochastically increasing family of measures. We prove that whenever the underlying particle configuration is sampled from an insertion-tolerant measure and the avalanches generated by the dynamics produce connected sets, the supercritical phase almost surely contains a unique infinite cluster. Our result applies to several well-studied interacting particle systems, including the Abelian sandpile, activated random walk, and bootstrap percolation. In these models, the induced percolation measure typically does not satisfy standard conditions such as finite energy or insertion tolerance, so the classical Burton-Keane argument does not apply. As an application, we answer a question of Fey, Meester, and Redig concerning percolation of the toppled vertices in the Abelian sandpile with i.i.d. initial configuration.

math.PR↗

arXiv2026

Percolation on graphs of polynomial growth is local: analyticity, supercritical sharpness, isoperimetry

We investigate locality of the supercritical regime for Bernoulli percolation on transitive graphs with polynomial growth, by which we mean the following. Take a transitive graph of polynomial growth $\mathscr{G}$ satisfying $p_c(\mathscr{G})<1$ and take $p>p_c(\mathscr{G})$. Let $\mathscr{H}$ be another such graph and assume that $\mathscr{G}$ and $\mathscr{H}$ have the same ball of radius $r$ for $r$ large. We prove that various quantities regarding percolation of parameter close to $p$ on $\mathscr{H}$ can be well understood from $(\mathscr{G},p)$ alone. This includes uniform versions of supercritical sharpness as well as the Kesten-Zhang bound on the probability of observing a large finite cluster: the constants involved can be chosen to depend only on $(\mathscr{G},p)$. We also prove that $θ_\mathscr{H}$ is an analytic function of $p$ in the whole supercritical regime and that, for a suitable $\varepsilon=\varepsilon(\mathscr{G},p)>0$, the analytic extension of $θ_\mathscr{H}$ to the $\varepsilon$-neighbourhood of $p$ in $\mathbb C$ is, uniformly, well approximated by the analytic extension of $θ_\mathscr{G}$. The proof relies on new results on the connectivity of minimal cutsets; in particular, we answer a question asked by Babson and Benjamini in 1999. We further discuss connections with the conjecture of non-percolation at criticality.

math.PR↗

arXiv2025

The supercritical phase of the $φ^4$ model is well behaved

In this article, we analyse the $φ^4$ model on $\mathbb Z^d$ in the supercritical regime $β> β_c$. We consider a random cluster representation of the $φ^4$ model, which corresponds to an Ising random cluster model on a random environment. We prove that the supercritical phase of this percolation model on $\mathbb Z^d$ ($d\geq 2$) is well behaved in the sense that, for every $β>β_c$, local uniqueness of macroscopic clusters occurs with high probability, uniformly in the boundary conditions. This result provides the basis for renormalisation techniques used to study several fine properties of the supercritical phase. As applications, we prove surface order exponential bounds for the (lower) large deviations of the empirical magnetisation as well as for the spectral gaps of dynamical $φ^4$ models in the entire supercritical regime.

math.PR↗

arXiv2025

Random tangled currents for $φ^4$: translation invariant Gibbs measures and continuity of the phase transition

We prove that the set of automorphism invariant Gibbs measures for the $φ^4$ model on graphs of polynomial growth has at most two extremal measures at all values of $β$. We also give a sufficient condition to ensure that the set of all Gibbs measures is a singleton. As an application, we show that the spontaneous magnetisation of the nearest-neighbour $φ^4$ model on $\mathbb{Z}^d$ vanishes at criticality for $d\geq 3$. The analogous results were established for the Ising model in the seminal works of Aizenman, Duminil-Copin, and Sidoravicius (Comm. Math. Phys., 2015), and Raoufi (Ann. Prob., 2020) using the so-called random current representation introduced by Aizenman (Comm. Math. Phys., 1982). One of the main contributions of this paper is the development of a corresponding geometric representation for the $φ^4$ model called the random tangled current representation.

math.PR↗

arXiv2024

Scaling limit of the cluster size distribution for the random current measure on the complete graph

We study the percolation configuration arising from the random current representation of the near-critical Ising model on the complete graph. We compute the scaling limit of the cluster size distribution for an arbitrary set of sources in the single and the double current measures. As a byproduct, we compute the tangling probabilities recently introduced by Gunaratnam, Panagiotis, Panis, and Severo in [GPPS22]. This provides a new perspective on the switching lemma for the $φ^4$ model introduced in the same paper: in the Gaussian limit we recover Wick's law, while in the Ising limit we recover the corresponding tool for the Ising model.

math.PR↗

arXiv2024

Existence of a tricritical point for the Blume-Capel model on $\mathbb{Z}^d$

We prove the existence of a tricritical point for the Blume-Capel model on $\mathbb{Z}^d$ for every $d\geq 2$. The proof in $d\geq 3$ relies on a novel combinatorial mapping to an Ising model on a larger graph, the techniques of Aizenman, Duminil-Copin, and Sidoravicious (Comm. Math. Phys, 2015), and the celebrated infrared bound. In $d=2$, the proof relies on a quantitative analysis of crossing probabilities of the dilute random cluster representation of the Blume-Capel. In particular, we develop a quadrichotomy result in the spirit of Duminil-Copin and Tassion (Moscow Math. J., 2020), which allows us to obtain a fine picture of the phase diagram in $d=2$, including asymptotic behaviour of correlations in all regions. Finally, we show that the techniques used to establish subcritical sharpness for the dilute random cluster model extend to any $d\geq 2$.

math.PR↗

arXiv2024

Self-avoiding walk is ballistic on graphs with more than one end

We prove that on any transitive graph $G$ with infinitely many ends, a self-avoiding walk of length $n$ is ballistic with extremely high probability, in the sense that there exist constants $c,t>0$ such that $\mathbb{P}_n(d_G(w_0,w_n)\geq cn)\geq 1-e^{-tn}$ for every $n\geq 1$. Furthermore, we show that the number of self-avoiding walks of length $n$ grows asymptotically like $μ_w^n$, in the sense that there exists $C>0$ such that $μ_w^n\leq c_n\leq Cμ_w^n$ for every $n\geq 1$. Our results extend more generally to quasi-transitive graphs with infinitely many ends, satisfying the additional technical property that there is a quasi-transitive group of automorphisms of $G$ which does not fix an end of $G$.

math.CO↗

arXiv2023

Quantitative sub-ballisticity of self-avoiding walk on the hexagonal lattice

We prove quantitative sub-ballisticity for the self-avoiding walk on the hexagonal lattice. Namely, we show that with high probability a self-avoiding walk of length $n$ does not exit a ball of radius $O(n/\log{n})$. Previously, only a non-quantitative $o(n)$ bound was known from the work of Duminil-Copin and Hammond \cite{DCH13}. As an important ingredient of the proof we show that at criticality the partition function of bridges of height $T$ decays polynomially fast to $0$ as $T$ tends to infinity, which we believe to be of independent interest.

math.PR↗

arXiv2022

Self-avoiding walks and polygons on hyperbolic graphs

We prove that for the $d$-regular tessellations of the hyperbolic plane by $k$-gons, there are exponentially more self-avoiding walks of length $n$ than there are self-avoiding polygons of length $n$. We then prove that this property implies that the self-avoiding walk is ballistic, even on an arbitrary vertex-transitive graph. Moreover, for every fixed $k$, we show that the connective constant for self-avoiding walks satisfies the asymptotic expansion $d-1-O(1/d)$ as $d\to \infty$; on the other hand, the connective constant for self-avoiding polygons remains bounded. Finally, we show for all but two tessellations that the number of self-avoiding walks of length $n$ is comparable to the $n$th power of their connective constant. Some of these results were previously obtained by Madras and Wu \cite{MaWuSAW} for all but finitely many regular tessellations of the hyperbolic plane.

math.PR↗

arXiv2021

Gap at 1 for the percolation threshold of Cayley graphs

We prove that the set of possible values for the percolation threshold $p_c$ of Cayley graphs has a gap at 1 in the sense that there exists $\varepsilon_0>0$ such that for every Cayley graph $G$ one either has $p_c(G)=1$ or $p_c(G) \leq 1-\varepsilon_0$. The proof builds on the new approach of Duminil-Copin, Goswami, Raoufi, Severo & Yadin to the existence of phase transition using the Gaussian free field, combined with the finitary version of Gromov's theorem on the structure of groups of polynomial growth of Breuillard, Green & Tao.

math.PR↗

arXiv2021

Analyticity of Gaussian free field percolation observables

We prove that cluster observables of level-sets of the Gaussian free field on the hypercubic lattice $\mathbb{Z}^d$, $d\geq3$, are analytic on the whole off-critical regime $\mathbb{R}\setminus\{h_*\}$. This result concerns in particular the percolation density function $θ(h)$ and the (truncated) susceptibility $χ(h)$. As an important step towards the proof, we show the exponential decay in probability for the capacity of a finite cluster for all $h\neq h_*$, which we believe to be a result of independent interest. We also discuss the case of general transient graphs.

math.PR↗

arXiv2021

Analyticity results in Bernoulli Percolation

We prove that for Bernoulli percolation on $\mathbb{Z}^d$, $d\geq 2$, 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 $p_c^{site} \leq 1/2$.

math.PR↗

arXiv2021

On the exponential growth rates of lattice animals and interfaces

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 $d\to \infty$. 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 $p\in (0,1)$.

math.PR↗