Mixing time for the solid-on-solid model

Martinelli, Fabio; Sinclair, Alistair

doi:10.1145/1536414.1536492

Cited by 8 publications

(27 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let us however mention that, for the monotone surface dynamics, under natural assumptions mentioned in Remark 1 below, one can apply [5, Theorem 3.1] to obtain a lower bound of order L 2 / log L for the mixing time. Similarly, for the SOS model it is known that the spectral gap is at most of order L −2 [12], which by standard inequalities directly implies a lower bound of order L 2 for T mix .…”

Section: Introductionmentioning

confidence: 91%

“…Due to the nonstrictly convex character of the interaction, even obtaining a diffusive spectral gap bound (of order L −2 ) for zero boundary conditions has been a long-standing open problem. The analysis of an auxiliary non-local dynamics, in the spirit of [18], plus a judicious use of the Peres-Winkler inequalities were recently combined to obtain a mixing time upper bound O(L 5/2 ) [12], while again the conjectured behavior is O(L 2 log L).…”

Section: Introductionmentioning

confidence: 99%

“…Note that, since the evolution tends to a profile with vanishing mean curvature, R t grows with time and becomes much larger than the initial value R 0 ≃ L when equilibrium is approached. One key input is to prove that the equilibration time within such mesoscopic regions is of the correct order O(ℓ 2 ): this result can be obtained following recent important progress in [5] for monotone surfaces and in [12] for SOS (both use the Peres-Winkler inequalities and Wilson's argument [18] in an essential way). The other key point, which is the main contribution of this work, is to show that the mesoscopic dynamics is dominated with high probability by the deterministic mean curvature evolution of a macroscopic smooth profile with the appropriate boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Mixing Times of Monotone Surfaces and SOS Interfaces: A Mean Curvature Approach

Caputo

Martinelli

Toninelli

2012

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

We consider stochastic spin-flip dynamics for: (i) monotone discrete surfaces in Z 3 with planar boundary height and (ii) the one-dimensional discrete Solid-on-Solid (SOS) model confined to a box. In both cases we show almost optimal bounds O(L 2 polylog(L)) for the mixing time of the chain, where L is the natural size of the system. The dynamics at a macroscopic scale should be described by a deterministic mean curvature motion such that each point of the surface feels a drift which tends to minimize the local surface tension [17]. Inspired by this heuristics, our approach consists in bounding the dynamics with an auxiliary one which, with very high probability, follows quite closely the deterministic mean curvature evolution. Key technical ingredients are monotonicity, coupling and an argument due to D. Wilson [18] in the framework of lozenge tiling Markov Chains. Our approach works equally well for both models despite the fact that their equilibrium maximal height fluctuations occur on very different scales (log L for monotone surfaces and √ L for the SOS model). Finally, combining techniques from kinetically constrained spin systems [2] together with the above mixing time result, we prove an almost diffusive lower bound of order 1/L 2 polylog(L) for the spectral gap of the SOS model with horizontal size L and unbounded heights.2000 Mathematics Subject Classification: 60K35, 82C20

show abstract

Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Mixing Times of Monotone Surfaces and SOS Interfaces: A Mean Curvature Approach

Caputo

Martinelli

Toninelli

2012

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

show abstract

“…First, the posterior distribution of a Bayesian model is usually a much more complex object than the highly structured distributions in statistical physics for which meaningful bounds on the Markov chain mixing times are often obtained (e.g. [6], [23], [21]). Second, the transition probabilities of the Markov chain are themselves stochastic, since they depend on the underlying data-generating process.…”

Section: Introductionmentioning

confidence: 99%

On the computational complexity of high-dimensional Bayesian variable selection

2016

View full text Add to dashboard Cite

We study the computational complexity of Markov chain Monte Carlo (MCMC) methods for high-dimensional Bayesian linear regression under sparsity constraints. We first show that a Bayesian approach can achieve variable-selection consistency under relatively mild conditions on the design matrix. We then demonstrate that the statistical criterion of posterior concentration need not imply the computational desideratum of rapid mixing of the MCMC algorithm. By introducing a truncated sparsity prior for variable selection, we provide a set of conditions that guarantee both variable-selection consistency and rapid mixing of a particular Metropolis-Hastings algorithm. The mixing time is linear in the number of covariates up to a logarithmic factor. Our proof controls the spectral gap of the Markov chain by constructing a canonical path ensemble that is inspired by the steps taken by greedy algorithms for variable selection.

show abstract

“…, L} 2 with zero boundary conditions, floor at zero and ceiling at n + with log L ≤ n + ≤ L satisfies e cL ≤ T MIX ≤ e (1/c)L . The exponentially large mixing time in (1.2) is in striking contrast with the rapid mixing displayed by Glauber dynamics of the (1 + 1)D SOS model [12,38]. When d = 1 it is known that the main driving effect is a meancurvature motion which induces a diffusive relaxation to equilibrium, with T MIX of order L 2 up to poly(log L) corrections.…”

mentioning

confidence: 93%

Dynamics of $(2+1)$-dimensional SOS surfaces above a wall: Slow mixing induced by entropic repulsion

Caputo¹,

Lubetzky²,

Martinelli³

et al. 2014

Ann. Probab.

Self Cite

View full text Add to dashboard Cite

We study the Glauber dynamics for the (2 + 1)D Solid-On-Solid model above a hard wall and below a far away ceiling, on an L × L box of Z 2 with zero boundary conditions, at large inverse-temperature β. It was shown by Bricmont, El Mellouki and Fröhlich [J. Stat. Phys. 42 (1986) 743-798] that the floor constraint induces an entropic repulsion effect which lifts the surface to an average height H ≍ (1/β) log L.As an essential step in understanding the effect of entropic repulsion on the Glauber dynamics we determine the equilibrium height H to within an additive constant: H = (1/4β) log L + O(1). We then show that starting from zero initial conditions the surface rises to its final height H through a sequence of metastable transitions between consecutive levels. The time for a transition from height h = aH, a ∈ (0, 1), to height h + 1 is roughly exp(cL a ) for some constant c > 0. In particular, the mixing time of the dynamics is exponentially large in L, that is, TMIX ≥ e cL . We also provide the matching upper bound TMIX ≤ e c ′ L , requiring a challenging analysis of the statistics of height contours at low temperature and new coupling ideas and techniques. Finally, to emphasize the role of entropic repulsion we show that without a floor constraint at height zero the mixing time is no longer exponentially large in L.

show abstract

Mixing time for the solid-on-solid model

Cited by 8 publications

References 24 publications

Mixing Times of Monotone Surfaces and SOS Interfaces: A Mean Curvature Approach

Mixing Times of Monotone Surfaces and SOS Interfaces: A Mean Curvature Approach

On the computational complexity of high-dimensional Bayesian variable selection

Dynamics of $(2+1)$-dimensional SOS surfaces above a wall: Slow mixing induced by entropic repulsion

Contact Info

Product

Resources

About