arXiv subjects

Fabio Martinelli

Publications and source records attributed to Fabio Martinelli.

At least 37 records · Page 2Linked to original sources

arXiv2016

Relaxation to equilibrium of generalized East processes on ${\mathbb{Z}}^d$: Renormalization group analysis and energy-entropy competition

We consider a class of kinetically constrained interacting particle systems on ${\mathbb{Z}}^d$ which play a key role in several heuristic qualitative and quantitative approaches to describe the complex behavior of glassy dynamics. With rate one and independently among the vertices of ${\mathbb{Z}}^d$, to each occupation variable $η_x\in\{0,1\}$ a new value is proposed by tossing a $(1-q)$-coin. If a certain local constraint is satisfied by the current configuration the proposed move is accepted, otherwise it is rejected. For $d=1$, the constraint requires that there is a vacancy at the vertex to the left of the updating vertex. In this case, the process is the well-known East process. On ${\mathbb{Z}}^2$, the West or the South neighbor of the updating vertex must contain a vacancy, similarly, in higher dimensions. Despite of their apparent simplicity, in the limit $q\searrow0$ of low vacancy density, corresponding to a low temperature physical setting, these processes feature a rather complicated dynamic behavior with hierarchical relaxation time scales, heterogeneity and universality. Using renormalization group ideas, we first show that the relaxation time on ${\mathbb{Z}}^d$ scales as the $1/d$-root of the relaxation time of the East process, confirming indications coming from massive numerical simulations. Next, we compute the relaxation time in finite boxes by carefully analyzing the subtle energy-entropy competition, using a multiscale analysis, capacity methods and an algorithmic construction. Our results establish dynamic heterogeneity and a dramatic dependence on the boundary conditions. Finally, we prove a rather strong anisotropy property of these processes: the creation of a new vacancy at a vertex $x$ out of an isolated one at the origin (a seed) may occur on (logarithmically) different time scales which heavily depend not only on the $\ell_1$-norm of $x$ but also on its direction.

math.PR↗

arXiv2015

On the probability of staying above a wall for the (2+1)-dimensional SOS model at low temperature

We obtain sharp asymptotics for the probability that the (2+1)-dimensional discrete SOS interface at low temperature is positive in a large region. For a square region $Λ$, both under the infinite volume measure and under the measure with zero boundary conditions around $Λ$, this probability turns out to behave like $\exp(-τ_β(0) L \log L )$, with $τ_β(0)$ the surface tension at zero tilt, also called step free energy, and $L$ the box side. This behavior is qualitatively different from the one found for continuous height massless gradient interface models.

math.PR↗

arXiv2015

Semiring-based Specification Approaches for Quantitative Security

Our goal is to provide different semiring-based formal tools for the specification of security requirements: we quantitatively enhance the open-system approach, according to which a system is partially specified. Therefore, we suppose the existence of an unknown and possibly malicious agent that interacts in parallel with the system. Two specification frameworks are designed along two different (but still related) lines. First, by comparing the behaviour of a system with the expected one, or by checking if such system satisfies some security requirements: we investigate a novel approximate behavioural-equivalence for comparing processes behaviour, thus extending the Generalised Non Deducibility on Composition (GNDC) approach with scores. As a second result, we equip a modal logic with semiring values with the purpose to have a weight related to the satisfaction of a formula that specifies some requested property. Finally, we generalise the classical partial model-checking function, and we name it as quantitative partial model-checking in such a way to point out the necessary and sufficient conditions that a system has to satisfy in order to be considered as secure, with respect to a fixed security/functionality threshold-value.

cs.LO↗

arXiv2015

Random lattice triangulations: Structure and algorithms

The paper concerns lattice triangulations, that is, triangulations of the integer points in a polygon in $\mathbb{R}^2$ whose vertices are also integer points. Lattice triangulations have been studied extensively both as geometric objects in their own right and by virtue of applications in algebraic geometry. Our focus is on random triangulations in which a triangulation $σ$ has weight $λ^{|σ|}$, where $λ$ is a positive real parameter, and $|σ|$ is the total length of the edges in $σ$. Empirically, this model exhibits a "phase transition" at $λ=1$ (corresponding to the uniform distribution): for $λ<1$ distant edges behave essentially independently, while for $λ>1$ very large regions of aligned edges appear. We substantiate this picture as follows. For $λ<1$ sufficiently small, we show that correlations between edges decay exponentially with distance (suitably defined), and also that the Glauber dynamics (a local Markov chain based on flipping edges) is rapidly mixing (in time polynomial in the number of edges in the triangulation). This dynamics has been proposed by several authors as an algorithm for generating random triangulations. By contrast, for $λ>1$ we show that the mixing time is exponential. These are apparently the first rigorous quantitative results on the structure and dynamics of random lattice triangulations.

math.PR↗

arXiv2015

Dynamics of Lattice Triangulations on Thin Rectangles

We consider random lattice triangulations of $n\times k$ rectangular regions with weight $λ^{|σ|}$ where $λ>0$ is a parameter and $|σ|$ denotes the total edge length of the triangulation. When $λ\in(0,1)$ and $k$ is fixed, we prove a tight upper bound of order $n^2$ for the mixing time of the edge-flip Glauber dynamics. Combined with the previously known lower bound of order $\exp(Ω(n^2))$ for $λ>1$ [3], this establishes the existence of a dynamical phase transition for thin rectangles with critical point at $λ=1$.

math.PR↗

arXiv2015

Mixing time and local exponential ergodicity of the East-like process in $\mathbf Z^d$

The East process, a well known reversible linear chain of spins, represents the prototype of a general class of interacting particle systems with constraints modeling the dynamics of real glasses. In this paper we consider a generalization of the East process living in the d-dimensional lattice and we establish new progresses on the out- of-equilibrium behavior. In particular we prove a form of (local) exponential ergodicity when the initial distribution is far from the stationary one and we prove that the mixing time in a finite box grows linearly in the side of the box.

math.PR↗

arXiv2014

Cutoff for the East process

The East process is a 1D kinetically constrained interacting particle system, introduced in the physics literature in the early 90's to model liquid-glass transitions. Spectral gap estimates of Aldous and Diaconis in 2002 imply that its mixing time on $L$ sites has order $L$. We complement that result and show cutoff with an $O(\sqrt{L})$-window. The main ingredient is an analysis of the front of the process (its rightmost zero in the setup where zeros facilitate updates to their right). One expects the front to advance as a biased random walk, whose normal fluctuations would imply cutoff with an $O(\sqrt{L})$-window. The law of the process behind the front plays a crucial role: Blondel showed that it converges to an invariant measure $ν$, on which very little is known. Here we obtain quantitative bounds on the speed of convergence to $ν$, finding that it is exponentially fast. We then derive that the increments of the front behave as a stationary mixing sequence of random variables, and a Stein-method based argument of Bolthausen ('82) implies a CLT for the location of the front, yielding the cutoff result. Finally, we supplement these results by a study of analogous kinetically constrained models on trees, again establishing cutoff, yet this time with an $O(1)$-window.

math.PR↗

arXiv2014

Multi-level pinning problems for random walks and self-avoiding lattice paths

We consider a generalization of the classical pinning problem for integer-valued random walks conditioned to stay non-negative. More specifically, we take pinning potentials of the form $\sum_{j\geq 0}ε_j N_j$, where $N_j$ is the number of visits to the state $j$ and $\{ε_j\}$ is a non-negative sequence. Partly motivated by similar problems for low-temperature contour models in statistical physics, we aim at finding a sharp characterization of the threshold of the wetting transition, especially in the regime where the variance $σ^2$ of the single step of the random walk is small. Our main result says that, for natural choices of the pinning sequence $\{ε_j\}$, localization (respectively delocalization) occurs if $σ^{-2}\sum_{ j\geq0}(j+1)ε_j\geqδ^{-1}$ (respectively $\le δ$), for some universal $δ<1$. Our finding is reminiscent of the classical Bargmann-Jost-Pais criteria for the absence of bound states for the radial Schrödinger equation. The core of the proof is a recursive argument to bound the free energy of the model. Our approach is rather robust, which allows us to obtain similar results in the case where the random walk trajectory is replaced by a self-avoiding path $γ$ in $\mathbb Z^2$ with weight $\exp(-β|γ|)$, $|γ|$ being the length of the path and $β>0$ a large enough parameter. This generalization is directly relevant for applications to the above mentioned contour models.

math.PR↗

arXiv2014

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

We study the Glauber dynamics for the $(2+1)\mathrm{D}$ Solid-On-Solid model above a hard wall and below a far away ceiling, on an $L\times L$ box of $\mathbb{Z}^2$ 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 $H\asymp(1/β)\log L$. 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 $h=aH$, $a\in(0,1)$, 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, $T_{\mathrm{MIX}}\geq e^{cL}$. We also provide the matching upper bound $T_{\mathrm{MIX}}\leq e^{c'L}$, 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 $L$.

math.PR↗

arXiv2014

The influence of dimension on the relaxation process of East-like models

We consider the relaxation process and the out-of-equilibrium dynamics of natural generalizations to arbitrary dimensions of the well known one dimensional East process. These facilitated models are supposed to catch some of the main features of the complex dynamics of fragile glasses. We focus on the low temperature regime (small density $c \approx e^{-β}$ of the facilitating sites). In the literature the relaxation process for the above models has been assumed to be quasi-one dimensional and, in particular, their equilibration time has been computed using the relaxation time of the East model ($d=1$) on the equilibrium length scale $L_c=(1/c)^{1/d}$ in $d$-dimension. This led to the derivation of a super-Arrhenius scaling for the relaxation time of the form $T_{\rm rel} \asymp \exp(β^2/d\log 2)$. In a companion paper, using mainly renormalization group ideas and electrical networks methods, we rigorously establish that instead $T_{\rm rel} \asymp \exp(β^2/2d\log 2)$, a result showing that the relaxation process cannot be quasi-one-dimensional. The above scaling sharply confirms previous MCAMC (Monte Carlo with Absorbing Markov Chains) simulations. Next we compute the relaxation time at finite and mesoscopic length scales, and show a dramatic dependence on the boundary conditions, yet another indication of key dimensional effects. Our final result is related to the out-of-equilibrium dynamics. Starting with a single facilitating site at the origin we show that, up to length scales $L=O(L_c)$, its influence propagates much faster (on a logarithmic scale) along the diagonal direction than along the axes directions. Such unexpected result is due to a rather delicate balance between dynamical energy barriers and entropic effects in the constrained dynamics.

cond-mat.stat-mech↗

arXiv2014

Harmonic pinnacles in the Discrete Gaussian model

The 2D Discrete Gaussian model gives each height function $η: \mathbb{Z}^2\to\mathbb{Z}$ a probability proportional to $\exp(-β\mathcal{H}(η))$, where $β$ is the inverse-temperature and $\mathcal{H}(η) = \sum_{x\sim y}(η_x-η_y)^2$ sums over nearest-neighbor bonds. We consider the model at large fixed $β$, where it is flat unlike its continuous analog (the Gaussian Free Field). We first establish that the maximum height in an $L\times L$ box with 0 boundary conditions concentrates on two integers $M,M+1$ with $M\sim \sqrt{(1/2πβ)\log L\log\log L}$. The key is a large deviation estimate for the height at the origin in $\mathbb{Z}^2$, dominated by "harmonic pinnacles", integer approximations of a harmonic variational problem. Second, in this model conditioned on $η\geq 0$ (a floor), the average height rises, and in fact the height of almost all sites concentrates on levels $H,H+1$ where $H\sim M/\sqrt{2}$. This in particular pins down the asymptotics, and corrects the order, in results of Bricmont, El-Mellouki and Fröhlich (1986), where it was argued that the maximum and the height of the surface above a floor are both of order $\sqrt{\log L}$. Finally, our methods extend to other classical surface models (e.g., restricted SOS), featuring connections to $p$-harmonic analysis and alternating sign matrices.

math.PR↗

arXiv2013

Time scale separation and dynamic heterogeneity in the low temperature East model

We consider the non-equilibrium dynamics of the East model, a linear chain of 0-1 spins evolving under a simple Glauber dynamics in the presence of a kinetic constraint which forbids flips of those spins whose left neighbor is 1. We focus on the glassy effects caused by the kinetic constraint as $q\downarrow 0$, where $q$ is the equilibrium density of the 0's. In the physical literature this limit is equivalent to the zero temperature limit. We first prove that, for any given $L=O(1/q)$, the divergence as $q\downarrow 0$ of three basic characteristic time scales of the East process of length $L$ is the same. Then we examine the problem of dynamic heterogeneity, i.e. non-trivial spatio-temporal fluctuations of the local relaxation to equilibrium, one of the central aspects of glassy dynamics. For any mesoscopic length scale $L=O(q^{-γ})$, $γ<1$, we show that the characteristic time scale of two East processes of length $L$ and $λL$ respectively are indeed separated by a factor $q^{-a}$, $a=a(γ)>0$, provided that $λ\geq 2$ is large enough (independent of $q$, $λ=2$ for $γ<1/2$). In particular, the evolution of mesoscopic domains, i.e. maximal blocks of the form $111..10$, occurs on a time scale which depends sharply on the size of the domain, a clear signature of dynamic heterogeneity. A key result for this part is a very precise computation of the relaxation time of the chain as a function of $(q,L)$, well beyond the current knowledge, which uses induction on length scales on one hand and a novel algorithmic lower bound on the other. Finally we show that no form of time scale separation occurs for $γ=1$, i.e. at the equilibrium scale $L=1/q$, contrary to what was assumed in the physical literature based on numerical simulations.

math-ph↗

arXiv2013

Mixing time bounds for oriented kinetically constrained spin models

We analyze the mixing time of a class of oriented kinetically constrained spin models (KCMs) on a d-dimensional lattice of $n^d$ sites. A typical example is the North-East model, a 0-1 spin system on the two-dimensional integer lattice that evolves according to the following rule: whenever a site's southerly and westerly nearest neighbours have spin 0, with rate one it resets its own spin by tossing a p-coin, at all other times its spin remains frozen. Such models are very popular in statistical physics because, in spite of their simplicity, they display some of the key features of the dynamics of real glasses. We prove that the mixing time is O(n log n) whenever the relaxation time is O(1). Our study was motivated by the "shape" conjecture put forward by G. Kordzakhia and S.P. Lalley.

math.PR↗

arXiv2013

Scaling limit and cube-root fluctuations in SOS surfaces above a wall

Consider the classical $(2+1)$-dimensional Solid-On-Solid model above a hard wall on an $L\times L$ box of $\bbZ^2$. The model describes a crystal surface by assigning a non-negative integer height $η_x$ to each site $x$ in the box and 0 heights to its boundary. The probability of a surface configuration $η$ is proportional to $\exp(-β\mathcal{H}(η))$, where $β$ is the inverse-temperature and $\mathcal{H}(η)$ sums the absolute values of height differences between neighboring sites. We give a full description of the shape of the SOS surface for low enough temperatures. First we show that with high probability the height of almost all sites is concentrated on two levels, $H(L)=\lfloor (1/4β)\log L\rfloor$ and $H(L)-1$. Moreover, for most values of $L$ the height is concentrated on the single value $H(L)$. Next, we study the ensemble of level lines corresponding to the heights $(H(L),H(L)-1,...)$. We prove that w.h.p. there is a unique macroscopic level line for each height. Furthermore, when taking a diverging sequence of system sizes $L_k$, the rescaled macroscopic level line at height $H(L_k)-n$ has a limiting shape if the fractional parts of $(1/4β)\log L_k$ converge to a noncritical value. The scaling limit is an explicit convex subset of the unit square $Q$ and its boundary has a flat component on the boundary of $Q$. Finally, the highest macroscopic level line has $L_k^{1/3+o(1)}$ fluctuations along the flat part of the boundary of its limiting shape.

math.PR↗

arXiv2013

Survey of Period Variations of Superhumps in SU UMa-Type Dwarf Novae. IV: The Fourth Year (2011-2012)

Continuing the project described by Kato et al. (2009, arXiv:0905.1757), we studied 86 SU UMa-type dwarf novae. We confirmed the general trends such as the relation between period derivatives and orbital periods. There are some systems showing positive period derivatives despite the long orbital periods. We observed the 2011 outburst of the WZ Sge-type dwarf nova BW Scl, and recorded an O-C diagram similar to those of previously known WZ Sge-type dwarf novae. The WZ Sge-type dwarf nova OT J184228.1+483742 showed an unusual pattern of double outbursts composed of an outburst with early superhumps and one with ordinary superhumps. We propose an interpretation that a very small growth rate of the 3:1 resonance due to an extremely low mass-ratio led to a quenching of the superoutburst before the ordinary superhumps appeared. We studied ER UMa-type dwarf novae and found that V1159 Ori showed positive superhumps similar to ER UMa in the 1990s. The recently recognized ER UMa-type object BK Lyn dominantly showed negative superhumps, and its behavior was very similar to the present-day state of ER UMa. The pattern of period variations in AM CVn-type objects were very similar to short-period hydrogen-rich SU UMa-type dwarf novae, making them helium analogue of hydrogen-rich SU UMa-type dwarf novae. SBS 1108+574, a peculiar hydrogen-rich dwarf nova below the period minimum, showed a very similar pattern of period variations to those of short-period SU UMa-type dwarf novae. The mass-ratio derived from the detected orbital period suggests that this secondary is a somewhat evolved star whose hydrogen envelope was mostly stripped during the mass-exchange. CC Scl, MASTER OT J072948.66+593824.4 and OT J173516.9+154708 showed only low-amplitude superhumps with complex profiles. These superhumps are likely a combination of closely separated two periods.

astro-ph.SR↗

arXiv2013

Time scale separation in the low temperature East model: Rigorous results

We consider the non-equilibrium dynamics of the East model, a linear chain of 0-1 spins evolving under a simple Glauber dynamics in the presence of a kinetic constraint which forbids flips of those spins whose left neighbour is 1. We focus on the glassy effects caused by the kinetic constraint as $q\downarrow 0$, where $q$ is the equilibrium density of the 0's. Specifically we analyse time scale separation and dynamic heterogeneity, i.e. non-trivial spatio-temporal fluctuations of the local relaxation to equilibrium, one of the central aspects of glassy dynamics. For any mesoscopic length scale $L=O(q^{-γ})$, $γ<1$, we show that the characteristic time scale associated to two length scales $d/q^γ$ and $d'/q^γ$ are indeed separated by a factor $q^{-a}$, $a=a(γ)>0$, provided that $d'/d$ is large enough independently of $q$. In particular, the evolution of mesoscopic domains, i.e. maximal blocks of the form $111..10$, occurs on a time scale which depends sharply on the size of the domain, a clear signature of dynamic heterogeneity. Finally we show that no form of time scale separation can occur for $γ=1$, i.e. at the equilibrium scale $L=1/q$, contrary to what was previously assumed in the physical literature based on numerical simulations.

cond-mat.stat-mech↗

arXiv2012

Metric-Aware Secure Service Orchestration

Secure orchestration is an important concern in the internet of service. Next to providing the required functionality the composite services must also provide a reasonable level of security in order to protect sensitive data. Thus, the orchestrator has a need to check whether the complex service is able to satisfy certain properties. Some properties are expressed with metrics for precise definition of requirements. Thus, the problem is to analyse the values of metrics for a complex business process. In this paper we extend our previous work on analysis of secure orchestration with quantifiable properties. We show how to define, verify and enforce quantitative security requirements in one framework with other security properties. The proposed approach should help to select the most suitable service architecture and guarantee fulfilment of the declared security requirements.

cs.CR↗

arXiv2012

Mixing time of a kinetically constrained spin model on trees: power law scaling at criticality

On the rooted $k$-ary tree we consider a 0-1 kinetically constrained spin model in which the occupancy variable at each node is re-sampled with rate one from the Bernoulli(p) measure iff all its children are empty. For this process the following picture was conjectured to hold. As long as $p$ is below the percolation threshold $p_c=1/k$ the process is ergodic with a finite relaxation time while, for $p>p_c$, the process on the infinite tree is no longer ergodic and the relaxation time on a finite regular sub-tree becomes exponentially large in the depth of the tree. At the critical point $p=p_c$ the process on the infinite tree is still ergodic but with an infinite relaxation time. Moreover, on finite sub-trees, the relaxation time grows polynomially in the depth of the tree. The conjecture was recently proved by the second and forth author except at criticality. Here we analyse the critical and quasi-critical case and prove for the relevant time scales: (i) power law behaviour in the depth of the tree at $p=p_c$ and (ii) power law scaling in $(p_c-p)^{-1}$ when $p$ approaches $p_c$ from below. Our results, which are very close to those obtained recently for the Ising model at the spin glass critical point, represent the first rigorous analysis of a kinetically constrained model at criticality.

math.PR↗