arXiv subjects

Sébastien Ott

Publications and source records attributed to Sébastien Ott.

At least 19 recordsLinked to original sources

arXiv2026

A new perspective on the equivalence between Weak and Strong Spatial Mixing in two dimensions

Weak mixing in lattice models is informally the property that ``information does not propagate inside a system''. Strong mixing is the property that ``information does not propagate inside and on the boundary of a system''. In dimension two, the boundary of reasonable systems is one dimensional, so information should not be able to propagate there. This led to the conjecture that in 2D, weak mixing implies strong mixing. The question was investigated in several previous works, and proof of this conjecture is available in the case of finite range Gibbsian specifications, and in the case of nearest-neighbour FK percolation (under some restrictions). The present work gives a new proof of these results, extends the family of models for which the implication holds, and, most interestingly, provides a ``percolative picture'' of the information propagation.

math.PR↗

arXiv2026

Discontinuous transition in 2D Potts: I. Order-Disorder Interface convergence

We study a $q$-state Potts model on the square grid when $q>4$ at the point $T_c(q)$ of its (discontinous) transition. This model exhibits exactly $q+1$ extremal Gibbs measures: $q$ ordered (monochromatic) and one disordered (free). The current work deals with the Dobrushin order--disorder boundary conditions on a finite $N\times N$ box. Our main result is that this interface is a well-defined object, has $\sqrt{N}$ fluctuations, and converges to a Brownian bridge under diffusive scaling. The same holds also for the corresponding FK-percolation model for all $q>4$. Our proofs rely on a coupling between FK-percolation, the six-vertex model, and the random-cluster representation of an Ashkin--Teller model (ATRC), and on a detailed study of the latter. The coupling relates the interface in FK-percolation to a long subcritical cluster in the ATRC model. For this cluster we develop a ``renewal picture'' à la Ornstein-Zernike. This is based on fine mixing properties of the ATRC model that we establish using the link to the six-vertex model and its height function. Along the way, we derive various properties of the Ashkin-Teller model, such as Ornstein-Zernike asymptotics for its two-point function. In a companion work, we provide a detailed study of the Potts model under order-order Dobrushin conditions. We show emergence of a free layer of width $\sqrt{N}$ between the two ordered phases (wetting) and establish convergence of its boundaries to two Brownian bridges conditioned not to intersect.

math.PR↗

arXiv2026

Discontinuous transition in 2D Potts: II. Order-Order Interface convergence

The $q$-state Potts model is an archetypical model for various types of phase transitions. We consider it on the square grid and focus on the regime where it undergoes a discontinuous transition, that is $q>4$. At the transition point $T_c(q)$, there are exactly $q+1$ extremal Gibbs measures (pure phases): $q$ ordered (monochromatic) and one disordered (free). This work establishes for the first time the wetting phenomenon in a precise geometric form and in the entire regime of discontinuity $q>4$: at $T_c(q)$, between two ordered phases a disordered layer emerges and, in the diffusive scaling, its boundaries converge to a pair of Brownian motions conditioned not to intersect. This is starkly different from the subcritical ($T 4$.

math.PR↗

arXiv2025

Finite-time trajectorial estimates for inhomogeneous random walks

We consider integer-valued random walks with independent but not identically distributed increments, and extend to this context several classical estimates, including a local limit theorem, precise small-ball estimates (both conditional on the final point and unconditional), and bounds on the probability that the random walk trajectory remains positive up to a given time (again, both conditional on the final point and unconditional). Two key features of this work are that the bounds are non-asymptotic, holding true for finite time horizons, and, crucially, that the latter hold uniformly over an entire class of admissible increment sequences. This provides a robust framework for applications. These results are, in particular, tailored for the analysis of processes derived through a time-dependent tilting of the increments of a time-homogeneous random walk.

math.PR↗

arXiv2025

Random Walk conditioned to stay above a non-flat floor: curvature effects

Let $h:[0,1]\to\mathbb{R}$ be $C^2$ and such that $\sup_{[0,1]} h''<0$. For a (large) positive integer $n$, set $h_n(k) = n h(k/n)$ for any $k\in\{0,\dots,n\}$. We consider a random walk $(S_k)_{k\geq 0}$ with i.i.d.\ centred increments having some finite exponential moments. We are interested in the event $\{S\geq h_n\} = \{S_k\geq h_n(k)\;\forall k\in\{0,\dots,n\}\}$. It is well known that $P(S\geq h_n \,|\, S_0=0,\, S_n=\lceil h_n(n) \rceil) = e^{n\int_0^1 I(h'(s)) \,ds + o(n)}$, where $I$ is the Legendre-Fenchel transform of the log-moment generating function associated to the increments. We first prove that the leading correction is of order $e^{-Θ(n^{1/3})}$. We then turn our attention to the conditional random walk measure $P^h_n = P(\cdot \,|\, S\geq h_n, S_0=0, S_n=\lceil h_n(n) \rceil)$. We prove that the one-point tails are of the form $\mathbb{P}_n^h (S_k \geq h_n(k) + t n^{1/3} ) = e^{-Θ(t^{3/2})}$ for all $t<n^β$ for any $β\in (0,1/6)$. Moreover, we prove that, for any $r\geq 1$, $E_n^h((S_k-h_n(k))^r) = Θ(n^{r/3})$ and $\mathrm{Var}_{P_n^h}(S_k) = Θ(n^{2/3})$, for all $k$ far enough from $0$ and $n$. In addition, we show that $\mathrm{Cov}_{P_n^h}(S_k,S_\ell) \leq e^{-O(|\ell-k|/n^{2/3})}$ for all $k,\ell$ not too close to $0$ and $n$.

math.PR↗

arXiv2025

Quantitative delocalization for solid-on-solid models at high temperature and arbitrary tilt

We study a family of integer-valued random interface models on the two-dimensional square lattice that include the solid-on-solid model and more generally $p$-SOS models for $0<p\le2$, and prove that at sufficiently high temperature the interface is delocalized logarithmically uniformly in the boundary data. Fröhlich and Spencer had studied the analogous problem with free boundary data, and our proof is based on their multi-scale argument, with various technical improvements.

math.PR↗

arXiv2024

Fixed-magnetization Ising model with a slowly varying magnetic field

The motivation for this paper is the analysis of the fixed-density Ising lattice gas in the presence of a gravitational field. This is a seen as a particular instance of an Ising model with a slowly varying magnetic field in the fixed magnetization ensemble. We first characterize the typical magnetization profiles in the regime in which the contribution of the magnetic field competes with the bulk energy term. We then discuss in more detail the particular case of a gravitational field and the arising interfacial phenomena. In particular, we identify the macroscopic profile and propose several conjectures concerning the interface appearing in the phase coexistence regime. The latter are supported by explicit computations in an effective model. Finally, we state some conjectures concerning equilibrium crystal shapes in the presence of a gravitational field, when the latter contributes to the energy only to surface order.

math.PR↗

arXiv2024

Relaxation time and topology in 1D $O(N)$ models

We discuss the relaxation time (inverse spectral gap) of the one dimensional $O(N)$ model, for all $N$ and with two types of boundary conditions. We see how its low temperature asymptotic behavior is affected by the topology. The combination of the space dimension, which here is always 1, the boundary condition (free or periodic), and the spin state $S^{N-1}$, determines the existence or absence of non-trivial homotopy classes in some discrete version. Such non-trivial topology reflects in bottlenecks of the dynamics, creating metastable states that the system exits at exponential times; while when only one homotopy class exists the relaxation time depends polynomially on the temperature. We prove in the one dimensional case that, indeed, the relaxation time is a proxy to the model's topological properties via the exponential/polynomial dependence on the temperature.

math.PR↗

arXiv2023

Asymptotics of correlations in the Ising model: a brief survey

We present a brief survey of rigorous results on the asymptotic behavior of correlations between two local functions as the distance between their support diverges, concentrating on the Ising model on $\mathbb{Z}^d$ with finite-range ferromagnetic interactions.

math.PR↗

arXiv2023

A note on the renormalization group approach to the Central Limit Theorem

Two proofs of the Central Limit Theorem using a renormalization group approach are presented. The first proof is conducted under a third moment assumption and shows that a suitable renormalization group map is a contraction over the space of probability measures with a third moment. The second proof uses Lyapunov stability and works under a second moment condition. These are by far not the most optimal proofs of the CLT, and the main interest of the proofs are their existence, the CLT being the simplest case in which a renormalization group argument should apply. None of the tools used in this note are new. Similar proofs are known amongst expert in limit theorems, but explicit references are not so easy to come by for non-experts in the field.

math.PR↗

arXiv2023

Delocalisation and absolute-value-FKG in the solid-on-solid model

The solid-on-solid model is a model of height functions, introduced to study the interface separating the $+$ and $-$ phase in the Ising model. The planar solid-on-solid model thus corresponds to the three-dimensional Ising model. Delocalisation of this model at high temperature and at zero slope was first derived by Fröhlich and Spencer, in parallel to proving the Berezinskii-Kosterlitz-Thouless phase transition. The first main result of this article consists of a simple, alternative proof of delocalisation of the solid-on-solid model. In fact, the argument is more general: it works on any planar graph -- not just the square lattice -- and implies that the interface delocalises at any slope rather than exclusively at the zero slope. The second main result, proved independently, is that the absolute value of the height function in this model satisfies the FKG lattice condition. This property is believed to be intimately linked to the (quantitative) understanding of delocalisation, given the recent successes in the context of the square ice and (more generally) the six-vertex model, and it has already been used elsewhere in a new proof of the BKT transition. The new FKG inequality is shown to hold true for both the solid-on-solid model as well as for the discrete Gaussian model, which in this article implies that the two notions of delocalisation, namely delocalisation in finite volume and delocalisation of shift-invariant Gibbs measures, coincide.

math.PR↗

arXiv2023

Low temperature asymptotic expansion for classical $O(N)$ vector models

We consider classical $O(N)$ vector models in dimension three and higher and investigate the nature of the low-temperature expansions for their multipoint spin correlations. We prove that such expansions define asymptotic series, and derive explicit estimates on the error terms associated with their finite order truncations. The result applies, in particular, to the spontaneous magnetization of the 3D Heisenberg model. The proof combines a priori bounds on the moments of the local spin observables, following from reflection positivity and the infrared bound, with an integration-by-parts method applied systematically to a suitable integral representation of the correlation functions. Our method generalizes an approach, proposed originally by Bricmont and collaborators [6] in the context of the rotator model, to the case of non-abelian symmetry and non-gradient observables.

math-ph↗

arXiv2022

On the two-point function of the Potts model in the saturation regime

We consider the Random-Cluster model on $\mathbb{Z}^d$ with interactions of infinite range of the form $J_x = ψ(x)\mathsf{e}^{-ρ(x)}$ with $ρ$ a norm on $\mathbb{Z}^d$ and $ψ$ a subexponential correction. We first provide an optimal criterion ensuring the existence of a nontrivial saturation regime (that is, the existence of $β_{\rm sat}(s)>0$ such that the inverse correlation length in the direction $s$ is constant on $[0,β_{\rm sat}(s))$), thus removing a regularity assumption used in a previous work of ours. Then, under suitable assumptions, we derive sharp asymptotics (which are not of Ornstein-Zernike form) for the two-point function in the whole saturation regime $(0,β_{\rm sat}(s))$. We also obtain a number of additional results for this class of models, including sharpness of the phase transition, mixing above the critical temperature and the strict monotonicity of the inverse correlation length in $β$ in the regime $(β_{\rm sat}(s), β_{\rm c})$.

math.PR↗

arXiv2022

Ornstein-Zernike behavior for Ising models with infinite-range interactions

We prove Ornstein-Zernike behavior for the large-distance asymptotics of the two-point function of the Ising model above the critical temperature under essentially optimal assumptions on the interaction. The main contribution of this work is that the interactions are not assumed to be of finite range. To the best of our knowledge, this is the first proof of OZ asymptotics for a nontrivial model with infinite-range interactions. Our results actually apply to the Green function of a large class of "self-repulsive in average" models, including a natural family of self-repulsive polymer models that contains, in particular, the self-avoiding walk, the Domb-Joyce model and the killed random walk. We aimed at a pedagogical and self-contained presentation.

math-ph↗

arXiv2021

Critical prewetting in the 2d Ising model

In this paper we develop a detailed analysis of critical prewetting in the context of the two-dimensional Ising model. Namely, we consider a two-dimensional nearest-neighbor Ising model in a $2N\times N$ rectangular box with a boundary condition inducing the coexistence of the $+$ phase in the bulk and a layer of $-$ phase along the bottom wall. The presence of an external magnetic field of intensity $h=λ/N$ (for some fixed $λ>0$) makes the layer of $-$ phase unstable. For any $β>β_{\rm c}$, we prove that, under a diffusing scaling by $N^{-2/3}$ horizontally and $N^{-1/3}$ vertically, the interface separating the layer of unstable phase from the bulk phase weakly converges to an explicit Ferrari-Spohn diffusion.

math.PR↗

arXiv2021

Non-analyticity of the correlation length in systems with exponentially decaying interactions

We consider a variety of lattice spin systems (including Ising, Potts and XY models) on $\mathbb{Z}^d$ with long-range interactions of the form $J_x = ψ(x) e^{-|x|}$, where $ψ(x) = e^{\mathsf{o}(|x|)}$ and $|\cdot|$ is an arbitrary norm. We characterize explicitly the prefactors $ψ$ that give rise to a correlation length that is not analytic in the relevant external parameter(s) (inverse temperature $β$, magnetic field $h$, etc). Our results apply in any dimension. As an interesting particular case, we prove that, in one-dimensional systems, the correlation length is non-analytic whenever $ψ$ is summable, in sharp contrast to the well-known analytic behavior of all standard thermodynamic quantities. We also point out that this non-analyticity, when present, also manifests itself in a qualitative change of behavior of the 2-point function. In particular, we relate the lack of analyticity of the correlation length to the failure of the mass gap condition in the Ornstein--Zernike theory of correlations.

math.PR↗

arXiv2020

Existence and properties of connections decay rate for high temperature percolation models

We consider generic finite range percolation models on $\mathbb{Z}^d$ under a high temperature assumption (exponential decay of connection probabilities and exponential ratio weak mixing). We prove that the rate of decay of point-to-point connections exists in every directions and show that it naturally extends to a norm on $\mathbb{R}^d$. This result is the base input to obtain fine understanding of the high temperature phase and is usually proven using correlation inequalities (such as FKG). The present work makes no use of such model specific properties.

math.PR↗

arXiv2020

Invariance principle for a Potts interface along a wall

We consider nearest-neighbor two-dimensional Potts models, with boundary conditions leading to the presence of an interface along the bottom wall of the box. We show that, after a suitable diffusive scaling, the interface weakly converges to the standard Brownian excursion.

math.PR↗