arXiv subjects

Reza Gheissari

Publications and source records attributed to Reza Gheissari.

46 records · Page 3Linked to original sources

arXiv2019

Random-cluster dynamics in $\mathbb Z^2$: rapid mixing with general boundary conditions

The random-cluster model with parameters $(p,q)$ is a random graph model that generalizes bond percolation ($q=1$) and the Ising and Potts models ($q\geq 2$). We study its Glauber dynamics on $n\times n$ boxes $Λ_{n}$ of the integer lattice graph $\mathbb Z^2$, where the model exhibits a sharp phase transition at $p=p_c(q)$. Unlike traditional spin systems like the Ising and Potts models, the random-cluster model has non-local interactions. Long-range interactions can be imposed as external connections in the boundary of $Λ_n$, known as boundary conditions. For select boundary conditions that do not carry long-range information (namely, wired and free), Blanca and Sinclair proved that when $q>1$ and $p\neq p_c(q)$, the Glauber dynamics on $Λ_n$ mixes in optimal $O(n^2 \log n)$ time. In this paper, we prove that this mixing time is polynomial in $n$ for every boundary condition that is realizable as a configuration on $\mathbb Z^2 \setminus Λ_{n}$. We then use this to prove near-optimal $\tilde O(n^2)$ mixing time for "typical'' boundary conditions. As a complementary result, we construct classes of non-realizable (non-planar) boundary conditions inducing slow (stretched-exponential) mixing at $p\ll p_c(q)$.

math.PR↗

arXiv2019

Quasi-polynomial mixing of critical 2D random cluster models

We study the Glauber dynamics for the random cluster (FK) model on the torus $(\mathbb{Z}/n\mathbb{Z})^2$ with parameters $(p,q)$, for $q \in (1,4]$ and $p$ the critical point $p_c$. The dynamics is believed to undergo a critical slowdown, with its continuous-time mixing time transitioning from $O(\log n)$ for $p\neq p_c$ to a power-law in $n$ at $p=p_c$. This was verified at $p\neq p_c$ by Blanca and Sinclair, whereas at the critical $p=p_c$, with the exception of the special integer points $q=2,3,4$ (where the model corresponds to the Ising/Potts models) the best-known upper bound on mixing was exponential in $n$. Here we prove an upper bound of $n^{O(\log n)}$ at $p=p_c$ for all $q\in (1,4]$, where a key ingredient is bounding the number of nested long-range crossings at criticality.

math.PR↗

arXiv2018

Ising Model: Local Spin Correlations and Conformal Invariance

We study the 2-dimensional Ising model at critical temperature on a simply connected subset $Ω_δ$ of the square grid $δ\mathbb{Z}^{2}$. The scaling limit of the critical Ising model is conjectured to be described by Conformal Field Theory; in particular, there is expected to be a precise correspondence between local lattice fields of the Ising model and the local fields of Conformal Field Theory. Towards the proof of this correspondence, we analyze arbitrary spin pattern probabilities (probabilities of finite spin configurations occurring at the origin), explicitly obtain their infinite-volume limits, and prove their conformal covariance at the first (non-trivial) order. We formulate these probabilities in terms of discrete fermionic observables, enabling the study of their scaling limits. This generalizes results of [Hon10,HoSm13] and [CHI15] to one-point functions of any local spin correlations. We introduce a collection of tools which allow one to exactly and explicitly translate any spin pattern probability (and hence any lattice local field correlation) in terms of discrete complex analysis quantities. The proof requires working with multipoint lattice spinors with monodromy (including construction of explicit formulae in the full plane), and refined analysis near their source points to prove convergence to the appropriate continuous conformally covariant functions.

math-ph↗

arXiv2018

The effect of boundary conditions on mixing of 2D Potts models at discontinuous phase transitions

We study Swendsen--Wang dynamics for the critical $q$-state Potts model on the square lattice. For $q=2,3,4$, where the phase transition is continuous, the mixing time $t_{\textrm{mix}}$ is expected to obey a universal power-law independent of the boundary conditions. On the other hand, for large $q$, where the phase transition is discontinuous, the authors recently showed that $t_{\textrm{mix}}$ is highly sensitive to boundary conditions: $t_{\textrm{mix}} \geq \exp(cn)$ on an $n\times n$ box with periodic boundary, yet under free or monochromatic boundary conditions, $t_{\textrm{mix}} \leq\exp(n^{o(1)})$. In this work we classify this effect under boundary conditions that interpolate between these two (torus vs. free/monochromatic). Specifically, if one of the $q$ colors is red, mixed boundary conditions such as red-free-red-free on the 4 sides of the box induce $t_{\textrm{mix}} \geq \exp(cn)$, yet Dobrushin boundary conditions such as red-red-free-free, as well as red-periodic-red-periodic, induce sub-exponential mixing.

math.PR↗

arXiv2018

On the Spectral Gap of Spherical Spin Glass Dynamics

We consider the time to equilibrium for the Langevin dynamics of the spherical $p$-spin glass model of system size $N$. We show that the log-Sobolev constant and spectral gap are order $1$ in $N$ at sufficiently high temperature whereas the spectral gap decays exponentially in $N$ at sufficiently low temperatures. These verify the existence of a dynamical high temperature phase and a dynamical glass phase at the level of the spectral gap. Key to these results are the understanding of the extremal process and restricted free energy of Subag--Zeitouni and Subag.

math.PR↗

arXiv2017

Concentration inequalities for polynomials of contracting Ising models

We study the concentration of a degree-$d$ polynomial of the $N$ spins of a general Ising model, in the regime where single-site Glauber dynamics is contracting. For $d=1$, Gaussian concentration was shown by Marton (1996) and Samson (2000) as a special case of concentration for convex Lipschitz functions, and extended to a variety of related settings by e.g., Chazottes et al. (2007) and Kontorovich and Ramanan (2008). For $d=2$, exponential concentration was shown by Marton (2003) on lattices. We treat a general fixed degree $d$ with $O(1)$ coefficients, and show that the polynomial has variance $O(N^d)$ and, after rescaling it by $N^{-d/2}$, its tail probabilities decay as $\exp(- c\, r^{2/d})$ for deviations of $r \geq C \log N$.

math.PR↗

arXiv2017

Mixing times of critical 2D Potts models

We study dynamical aspects of the $q$-state Potts model on an $n\times n$ box at its critical $β_c(q)$. Heat-bath Glauber dynamics and cluster dynamics such as Swendsen--Wang (that circumvent low-temperature bottlenecks) are all expected to undergo "critical slowdowns" in the presence of periodic boundary conditions: the inverse spectral gap, which in the subcritical regime is $O(1)$, should at criticality be polynomial in $n$ for $1< q \leq 4$, and exponential in $n$ for $q>4$ in accordance with the predicted discontinuous phase transition. This was confirmed for $q=2$ (the Ising model) by the second author and Sly, and for sufficiently large $q$ by Borgs et al. Here we show that the following holds for the critical Potts model on the torus: for $q=3$, the inverse gap of Glauber dynamics is $n^{O(1)}$; for $q=4$, it is at most $n^{O(\log n)}$; and for every $q>4$ in the phase-coexistence regime, the inverse gaps of both Glauber dynamics and Swendsen--Wang dynamics are exponential in $n$. For free or monochromatic boundary conditions and large $q$, we show that the dynamics at criticality is faster than on the torus (unlike the Ising model where free/periodic boundary conditions induce similar dynamical behavior at all temperatures): the inverse gap of Swendsen--Wang dynamics is $\exp(n^{o(1)})$.

math.PR↗

arXiv2017

Zero-temperature dynamics in the dilute Curie-Weiss model

We consider the Ising model on a dense Erdős--Rényi random graph, $\mathcal G(N,p)$, with $p>0$ fixed---equivalently, a disordered Curie--Weiss Ising model with $\mbox{Ber}(p)$ couplings---at zero temperature. The disorder may induce local energy minima in addition to the two uniform ground states. In this paper we prove that, starting from a typical initial configuration, the zero-temperature dynamics avoids all such local minima and absorbs into a predetermined one of the two uniform ground states. We relate this to the local MINCUT problem on dense random graphs; namely with high probability, the greedy search for a local MINCUT of $\mathcal G(N,p)$ with $p>0$ fixed, started from a uniform random partition, fails to find a non-trivial cut. In contrast, in the disordered Curie--Weiss model with heavy-tailed couplings, we demonstrate that zero-temperature dynamics has positive probability of absorbing in a random local minimum different from the two homogenous ground states.

math.PR↗

arXiv2017

Exponentially slow mixing in the mean-field Swendsen-Wang dynamics

Swendsen-Wang dynamics for the Potts model was proposed in the late 1980's as an alternative to single-site heat-bath dynamics, in which global updates allow this MCMC sampler to switch between metastable states and ideally mix faster. Gore and Jerrum (1999) found that this dynamics may in fact exhibit slow mixing: they showed that, for the Potts model with $q\geq 3$ colors on the complete graph on $n$ vertices at the critical point $β_c(q)$, Swendsen-Wang dynamics has $t_{\mathrm{mix}}\geq \exp(c\sqrt n)$. The same lower bound was extended to the critical window $(β_s,β_S)$ around $β_c$ by Galanis et al. (2015), as well as to the corresponding mean-field FK model by Blanca and Sinclair (2015). In both cases, an upper bound of $t_{\mathrm{mix}} \leq \exp(c' n)$ was known. Here we show that the mixing time is truly exponential in $n$: namely, $t_{\mathrm{mix}} \geq \exp (cn)$ for Swendsen-Wang dynamics when $q\geq 3$ and $β\in(β_s,β_S)$, and the same bound holds for the related MCMC samplers for the mean-field FK model when $q>2$.

math.PR↗

arXiv2014

Asymptotics of height change on toroidal Temperleyan dimer models

The dimer model is an exactly solvable model of planar statistical mechanics. In its critical phase, various aspects of its scaling limit are known to be described by the Gaussian free field. For periodic graphs, criticality is an algebraic condition on the spectral curve of the model, determined by the edge weights; isoradial graphs provide another class of critical dimer models, in which the edge weights are determined by the local geometry. In the present article, we consider another class of graphs: general Temperleyan graphs, i.e. graphs arising in the (generalized) Temperley bijection between spanning trees and dimer models. Building in particular on Forman's formula and representations of Laplacian determinants in terms of Poisson operators, and under a minimal assumption - viz. that the underlying random walk converges to Brownian motion - we show that the natural topological observable on macroscopic tori converges in law to its universal limit, i.e. the law of the periods of the dimer height function converges to that of the periods of a compactified free field.

math.PR↗