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arXiv2014

Statistical physics of hard combinatorial optimization: The vertex cover problem

Typical-case computation complexity is a research topic at the boundary of computer science, applied mathematics, and statistical physics. In the last twenty years the replica-symmetry-breaking mean field theory of spin glasses and the associated message-passing algorithms have greatly deepened our understanding of typical-case computation complexity. In this paper we use the vertex cover problem, a basic nondeterministic-polynomial (NP)-complete combinatorial optimization problem of wide application, as an example to introduce the statistical physical methods and algorithms. We do not go into the technical details but emphasize mainly the intuitive physical meanings of the message-passing equations. A nonfamiliar reader shall be able to understand to a large extent the physics behind the mean field approaches and to adjust them in solving other optimization problems.

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arXiv2012

Next nearest neighbour Ising models on random graphs

This paper develops results for the next nearest neighbour Ising model on random graphs. Besides being an essential ingredient in classic models for frustrated systems, second neighbour interactions interactions arise naturally in several applications such as the colour diversity problem and graphical games. We demonstrate ensembles of random graphs, including regular connectivity graphs, that have a periodic variation of free energy, with either the ratio of nearest to next nearest couplings, or the mean number of nearest neighbours. When the coupling ratio is integer paramagnetic phases can be found at zero temperature. This is shown to be related to the locked or unlocked nature of the interactions. For anti-ferromagnetic couplings, spin glass phases are demonstrated at low temperature. The interaction structure is formulated as a factor graph, the solution on a tree is developed. The replica symmetric and energetic one-step replica symmetry breaking solution is developed using the cavity method. We calculate within these frameworks the phase diagram and demonstrate the existence of dynamical transitions at zero temperature for cases of anti-ferromagnetic coupling on regular and inhomogeneous random graphs.

cond-mat.stat-mech↗

arXiv2014

Statistical mechanics approach to 1-bit compressed sensing

Compressed sensing is a technique for recovering a high-dimensional signal from lower-dimensional data, whose components represent partial information about the signal, utilizing prior knowledge on the sparsity of the signal. For further reducing the data size of the compressed expression, a scheme to recover the original signal utilizing only the sign of each entry of the linearly transformed vector was recently proposed. This approach is often termed the 1-bit compressed sensing. Here we analyze the typical performance of an L1-norm based signal recovery scheme for the 1-bit compressed sensing using statistical mechanics methods. We show that the signal recovery performance predicted by the replica method under the replica symmetric ansatz, which turns out to be locally unstable for modes breaking the replica symmetry, is in a good consistency with experimental results of an approximate recovery algorithm developed earlier. This suggests that the L1-based recovery problem typically has many local optima of a similar recovery accuracy, which can be achieved by the approximate algorithm. We also develop another approximate recovery algorithm inspired by the cavity method. Numerical experiments show that when the density of nonzero entries in the original signal is relatively large the new algorithm offers better performance than the abovementioned scheme and does so with a lower computational cost.

cs.IT↗

arXiv2013

The effect of quenched disorder in neutral theories

We study systems with two symmetric absorbing states, such as the voter model and variations of it, which have been broadly used as minimal neutral models in genetics, population ecology, sociology, etc. We analyze the effects of a key ingredient ineluctably present in most real applications: random-field-like quenched disorder. In accord with simulations and previous findings, coexistence between the two competing states/opinions turns out to be strongly favored by disorder in the standard voter model; actually, a disorder-induced phase transition is generated for any finite system size in the presence of an arbitrary small spontaneous-inversion rate (preventing absorbing states from being stable). For non-linear versions of the voter model a general theory (by AlHammal et al.) explains that the spontaneous breaking of the up/down symmetry and an absorbing state phase transition can occur either together or separately, giving raise to two different scenarios. Here, we show that he presence of quenched disorder in non-linear voter models does not allow the separation of the up-down (Ising-like) symmetry breaking from the active-to-absorbing phase transition in low-dimensional systems: both phenomena can occur only simultaneously, as a consequence of the well-known Imry-Ma argument generalized to these non-equilibrium problems. When the two phenomena occur at unison, resulting into a genuinely non-equilibrium ("Generalized Voter") transition, the Imry-Ma argument is violated and the symmetry can be spontaneously broken even in low dimensions.

cond-mat.stat-mech↗

arXiv2013

Dynamical barriers of pure and random ferromagnetic Ising models on fractal lattices

We consider the stochastic dynamics of the pure and random ferromagnetic Ising model on the hierarchical diamond lattice of branching ratio $K$ with fractal dimension $d_f=(\ln (2K))/\ln 2$. We adapt the Real Space Renormalization procedure introduced in our previous work [C. Monthus and T. Garel, J. Stat. Mech. P02037 (2013)] to study the equilibrium time $t_{eq}(L)$ as a function of the system size $L$ near zero-temperature. For the pure Ising model, we obtain the behavior $t_{eq}(L) \sim L^α e^{β2J L^{d_s}} $ where $d_s=d_f-1$ is the interface dimension, and we compute the prefactor exponent $α$. For the random ferromagnetic Ising model, we derive the renormalization rules for dynamical barriers $B_{eq}(L) \equiv (\ln t_{eq}/β)$ near zero temperature. For the fractal dimension $d_f=2$, we obtain that the dynamical barrier scales as $ B_{eq}(L)= c L+L^{1/2} u$ where $u$ is a Gaussian random variable of non-zero-mean. While the non-random term scaling as $L$ corresponds to the energy-cost of the creation of a system-size domain-wall, the fluctuation part scaling as $L^{1/2}$ characterizes the barriers for the motion of the system-size domain-wall after its creation. This scaling corresponds to the dynamical exponent $ψ=1/2$, in agreement with the conjecture $ψ=d_s/2$ proposed in [C. Monthus and T. Garel, J. Phys. A 41, 115002 (2008)]. In particular, it is clearly different from the droplet exponent $θ\simeq 0.299$ involved in the statics of the random ferromagnet on the same lattice.

cond-mat.dis-nn↗

arXiv2013

Two-time free energy distribution function in (1+1) directed polymers

The explicit expression for the two-time free energy distribution function in one-dimensional directed polymers in random potential is derived in terms of the Bethe ansatz replica technique by mapping the replicated problem to the $N$-particle quantum boson system with attractive interactions.

cond-mat.stat-mech↗

arXiv2013

Area laws in a many-body localized state and its implications for topological order

The question whether Anderson insulators can persist to finite-strength interactions - a scenario dubbed many-body localization - has recently received a great deal of interest. The origin of such a many-body localized phase has been described as localization in Fock space, a picture we examine numerically. We then formulate a precise sense in which a single energy eigenstate of a Hamiltonian can be adiabatically connected to a state of a non-interacting Anderson insulator. We call such a state a many-body localized state and define a many-body localized phase as one in which almost all states are many-body localized states. We explore the possible consequences of this; the most striking is an area law for the entanglement entropy of almost all excited states in a many-body localized phase. We present the results of numerical calculations for a one-dimensional system of spinless fermions. Our results are consistent with an area law and, by implication, many-body localization for almost all states and almost all regions for weak enough interactions and strong disorder. However, there are rare regions and rare states with much larger entanglement entropies. Furthermore, we study the implications that many-body localization may have for topological phases and self-correcting quantum memories. We find that there are scenarios in which many-body localization can help to stabilize topological order at non-zero energy density, and we propose potentially useful criteria to confirm these scenarios.

cond-mat.dis-nn↗

arXiv2013

Optimizing spread dynamics on graphs by message passing

Cascade processes are responsible for many important phenomena in natural and social sciences. Simple models of irreversible dynamics on graphs, in which nodes activate depending on the state of their neighbors, have been successfully applied to describe cascades in a large variety of contexts. Over the last decades, many efforts have been devoted to understand the typical behaviour of the cascades arising from initial conditions extracted at random from some given ensemble. However, the problem of optimizing the trajectory of the system, i.e. of identifying appropriate initial conditions to maximize (or minimize) the final number of active nodes, is still considered to be practically intractable, with the only exception of models that satisfy a sort of diminishing returns property called submodularity. Submodular models can be approximately solved by means of greedy strategies, but by definition they lack cooperative characteristics which are fundamental in many real systems. Here we introduce an efficient algorithm based on statistical physics for the optimization of trajectories in cascade processes on graphs. We show that for a wide class of irreversible dynamics, even in the absence of submodularity, the spread optimization problem can be solved efficiently on large networks. Analytic and algorithmic results on random graphs are complemented by the solution of the spread maximization problem on a real-world network (the Epinions consumer reviews network).

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arXiv2013

Uniqueness of the thermodynamic limit for driven disordered elastic interfaces

We study the finite size fluctuations at the depinning transition for a one-dimensional elastic interface of size $L$ displacing in a disordered medium of transverse size $M=k L^ζ$ with periodic boundary conditions, where $ζ$ is the depinning roughness exponent and $k$ is a finite aspect ratio parameter. We focus on the crossover from the infinitely narrow ($k\to 0$) to the infinitely wide ($k\to \infty$) medium. We find that at the thermodynamic limit both the value of the critical force and the precise behavior of the velocity-force characteristics are {\it unique} and $k$-independent. We also show that the finite size fluctuations of the critical force (bias and variance) as well as the global width of the interface cross over from a power-law to a logarithm as a function of $k$. Our results are relevant for understanding anisotropic size-effects in force-driven and velocity-driven interfaces.

cond-mat.stat-mech↗

arXiv2014

Calculation of 1RSB transition temperature of spin glass models on regular random graphs under the replica symmetric ansatz

We study $p$-spin glass models on regular random graphs. By analyzing the Franz-Parisi potential with a two-body cavity field approximation under the replica symmetric ansatz, we obtain a good approximation of the 1RSB transition temperature for $p=3$. Our calculation method is much easier than the 1RSB cavity method because the result is obtained by solving self-consistent equations with Newton's method.

cond-mat.stat-mech↗

arXiv2014

On the phase transition of the 3D random field Ising model

We present numerical simulations of the random field Ising model in three dimensions at zero temperature. The critical exponents are found to agree with previous results. We study the magnetic susceptibility by applying a small magnetic field perturbation. We find that the critical amplitude ratio of the magnetic susceptibilities to be very large, equal to 233.1 \pm 1.5. We find strong sample to sample fluctuations which obey finite size scaling. The probability distribution of the size of small energy excitations is maximally non-self averaging, obeying a double peak distribution, and is finite size scaling invariant. We also study the approach to the thermodynamic limit of the ground state magnetization at the phase transition.

cond-mat.dis-nn↗

arXiv2013

Chaos properties of the one-dimensional long-range Ising spin-glass

For the long-range one-dimensional Ising spin-glass with random couplings decaying as $J(r) \propto r^{-σ}$, the scaling of the effective coupling defined as the difference between the free-energies corresponding to Periodic and Antiperiodic boundary conditions $J^R(N) \equiv F^{(P)}(N)-F^{(AP)}(N) \sim N^{θ(σ)}$ defines the droplet exponent $θ(σ)$. Here we study numerically the instability of the renormalization flow of the effective coupling $J^R(N)$ with respect to magnetic, disorder and temperature perturbations respectively, in order to extract the corresponding chaos exponents $ζ_H(σ)$, $ζ_J(σ)$ and $ζ_T(σ)$ as a function of $σ$. Our results for $ζ_T(σ) $ are interpreted in terms of the entropy exponent $θ_S(σ) \simeq 1/3$ which governs the scaling of the entropy difference $ S^{(P)}(N)-S^{(AP)}(N) \sim N^{θ_S(σ)}$. We also study the instability of the ground state configuration with respect to perturbations, as measured by the spin overlap between the unperturbed and the perturbed ground states, in order to extract the corresponding chaos exponents $ζ^{overlap}_H(σ)$ and $ζ^{overlap}_J(σ)$.

cond-mat.dis-nn↗

arXiv2014

Rare events analysis of temperature chaos in the Sherrington-Kirkpatrick model

We investigate the question of temperature chaos in the Sherrington-Kirkpatrick spin glass model, applying to existing Monte Carlo data a recently proposed rare events based data analysis method. Thanks to this new method, temperature chaos is now observable for this model, even with the limited size systems that can be currently simulated.

cond-mat.dis-nn↗

arXiv2014

Interaction quench in a Lieb-Liniger model and the KPZ equation with flat initial conditions

Recent exact solutions of the 1D Kardar-Parisi-Zhang equation make use of the 1D integrable Lieb-Liniger model of interacting bosons. For flat initial conditions, it requires the knowledge of the overlap between the uniform state and arbitrary exact Bethe eigenstates. The same quantity is also central in the study of the quantum quench from a 1D non-interacting Bose-Einstein condensate upon turning interactions. We compare recent advances in both domains, i.e. our previous exact solution, and a new conjecture by De Nardis et al.. This leads to new exact results and conjectures for both the quantum quench and the KPZ problem.

cond-mat.dis-nn↗

arXiv2014

The three dimensional Ising spin glass in an external magnetic field: the role of the silent majority

We perform equilibrium parallel-tempering simulations of the 3D Ising Edwards-Anderson spin glass in a field. A traditional analysis shows no signs of a phase transition. Yet, we encounter dramatic fluctuations in the behaviour of the model: Averages over all the data only describe the behaviour of a small fraction of it. Therefore we develop a new approach to study the equilibrium behaviour of the system, by classifying the measurements as a function of a conditioning variate. We propose a finite-size scaling analysis based on the probability distribution function of the conditioning variate, which may accelerate the convergence to the thermodynamic limit. In this way, we find a non-trivial spectrum of behaviours, where a part of the measurements behaves as the average, while the majority of them shows signs of scale invariance. As a result, we can estimate the temperature interval where the phase transition in a field ought to lie, if it exists. Although this would-be critical regime is unreachable with present resources, the numerical challenge is finally well posed.

cond-mat.dis-nn↗

arXiv2014

Dynamics of asymmetric kinetic Ising systems revisited

The dynamics of an asymmetric kinetic Ising model is studied. Two schemes for improving the existing mean-field description are proposed. In the first scheme, we derive the formulas for instantaneous magnetization, equal-time correlation, and time-delayed correlation, considering the correlation between different local fields. To derive the time-delayed correlation, we emphasize that the small correlation assumption adopted in previous work [M. Mézard and J. Sakellariou, J. Stat. Mech., L07001 (2011)] is in fact not required. To confirm the inference efficiency of our method, we perform extensive simulations on single instances with either temporally constant external driving fields or sinusoidal external fields. In the second scheme, we develop an improved mean-field theory for instantaneous magnetization prediction utilizing the notion of the cavity system in conjunction with a perturbative expansion approach. Its efficiency is numerically confirmed by comparison with the existing mean-field theory when partially asymmetric couplings are present.

cond-mat.dis-nn↗

arXiv2014

Shortest node-disjoint paths on random graphs

A localized method to distribute paths on random graphs is devised, aimed at finding the shortest paths between given source/destination pairs while avoiding path overlaps at nodes. We propose a method based on message-passing techniques to process global information and distribute paths optimally. Statistical properties such as scaling with system size and number of paths, average path-length and the transition to the frustrated regime are analysed. The performance of the suggested algorithm is evaluated through a comparison against a greedy algorithm.

cond-mat.dis-nn↗

arXiv2014

On the Gap and Time Interval between the First Two Maxima of Long Random Walks

In the context of order statistics of discrete time random walks (RW), we investigate the statistics of the gap, $G_n$, and the number of time steps, $L_n$, between the two highest positions of a Markovian one-dimensional random walker, starting from $x_0 = 0$, after $n$ time steps (taking the $x$-axis vertical). The jumps $η_i = x_i - x_{i-1}$ are independent and identically distributed random variables drawn from a symmetric probability distribution function (PDF), $f(η)$, the Fourier transform of which has the small $k$ behavior $1 - \hat f(k) \propto |k|^μ$, with $0 < μ\leq 2$. For $μ=2$, the variance of the jump distribution is finite and the RW (properly scaled) converges to a Brownian motion. For $0<μ<2$, the RW is a Lévy flight of index $μ$. We show that the joint PDF of $G_n$ and $L_n$ converges to a well defined stationary bi-variate distribution $p(g,l)$ as the RW duration $n$ goes to infinity. We present a thorough analytical study of the limiting joint distribution $p(g,l)$, as well as of its associated marginals $p_{\rm gap}(g)$ and $p_{\rm time}(l)$, revealing a rich variety of behaviors depending on the tail of $f(η)$ (from slow decreasing algebraic tail to fast decreasing super-exponential tail). We also address the problem for a random bridge where the RW starts and ends at the origin after $n$ time steps. We show that in the large $n$ limit, the PDF of $G_n$ and $L_n$ converges to the {\it same} stationary distribution $p(g,l)$ as in the case of the free-end RW. Finally, we present a numerical check of our analytical predictions. Some of these results were announced in a recent letter [S. N. Majumdar, Ph. Mounaix, G. Schehr, Phys. Rev. Lett. {\bf 111}, 070601 (2013)].

cond-mat.stat-mech↗