Explore the vast range of titles available.

We prove that for Bernoulli percolation on

The Airy line ensemble is a positive-integer indexed system of random continuous curves whose finite dimensional distributions are given by the multi-line Airy process. It is a natural object in the KPZ universality class: for example, its highest curve, the Airy In this paper, we employ the Brownian Gibbs property to make a close comparison between the Airy line ensemble’s curves after affine shift and Brownian bridge, proving the finiteness of a superpolynomially growing moment bound on Radon-Nikodym derivatives. We also determine the value of a natural exponent describing in Brownian last passage percolation the decay in probability for the existence of several near geodesics that are disjoint except for their common endpoints, where the notion of ‘near’ refers to a small deficit in scaled geodesic energy, with the parameter specifying this nearness tending to zero. To prove both results, we introduce a technique that may be useful elsewhere for finding upper bounds on probabilities of events concerning random systems of curves enjoying the Brownian Gibbs property. Several results in this article play a fundamental role in a further study of Brownian last passage percolation in three companion papers (Hammond 2017a,b,c), in which geodesic coalescence and geodesic energy profiles are investigated in scaled coordinates.

Hard particle systems are among the most successful, inspiring and prolific models of physics. They contain the essential ingredients to understand a large class of complex phenomena, from phase transitions to glassy dynamics, jamming, or the physics of liquid crystals and granular materials, to mention just a few. As we discuss in this paper, their study also provides crucial insights on the problem of transport out of equilibrium. A main tool in this endeavour are computer simulations of hard particles. Here we review some of our work in this direction, focusing on the hard disks fluid as a model system. In this quest we will address, using extensive numerical simulations, some of the key open problems in the physics of transport, ranging from local equilibrium and Fourier’s law to the transition to convective flow in the presence of gravity, the efficiency of boundary dissipation, or the universality of anomalous transport in low dimensions. In particular, we probe numerically the macroscopic local equilibrium hypothesis, which allows to measure the fluid’s equation of state in nonequilibrium simulations, uncovering along the way subtle nonlocal corrections to local equilibrium and a remarkable bulk-boundary decoupling phenomenon in fluids out of equilibrium. We further show that the the hydrodynamic profiles that a system develops when driven out of equilibrium by an arbitrary temperature gradient obey universal scaling laws, a result that allows the determination of transport coefficients with unprecedented precission and proves that Fourier’s law remains valid in highly nonlinear regimes. Switching on a gravity field against the temperature gradient, we investigate numerically the transition to convective flow. We uncover a surprising two-step transition scenario with two different critical thresholds for the hot bath temperature, a first one where convection kicks but gravity hinders heat transport, and a second critical temperature where a percolation transition of streamlines connecting the hot and cold baths triggers efficient convective heat transport. We also address numerically the efficiency of boundary heat baths to dissipate the energy provided by a bulk driving mechanism. As a bonus track, we depart from the hard disks model to study anomalous transport in a related hard-particle system, the 1 d diatomic hard-point gas. We show unambiguously that the universality conjectured for anomalous transport in 1 d breaks down for this model, calling into question recent theoretical predictions and offering a new perspective on anomalous transport in low dimensions. Our results show how carefully-crafted numerical simulations of simple hard particle systems can lead to unexpected discoveries in the physics of transport, paving the way to further advances in nonequilibrium physics.

We define dynamical universality classes for many-body systems whose unitary evolution is punctuated by projective measurements. In cases where such measurements occur randomly at a finite ratepfor each degree of freedom, we show that the system has two dynamical phases: “entangling” and “disentangling.” The former occurs forpsmaller than a critical ratepcand is characterized by volume-law entanglement in the steady state and “ballistic” entanglement growth after a quench. By contrast, forp>pcthe system can sustain only area-law entanglement. Atp=pcthe steady state is scale invariant, and in1+1D, the entanglement grows logarithmically after a quench. To obtain a simple heuristic picture for the entangling-disentangling transition, we first construct a toy model that describes the zeroth Rényi entropy in discrete time. We solve this model exactly by mapping it to an optimization problem in classical percolation. The generic entangling-disentangling transition can be diagnosed using the von Neumann entropy and higher Rényi entropies, and it shares many qualitative features with the toy problem. We study the generic transition numerically in quantum spin chains and show that the phenomenology of the two phases is similar to that of the toy model but with distinct “quantum” critical exponents, which we calculate numerically in1+1D. We examine two different cases for the unitary dynamics: Floquet dynamics for a nonintegrable Ising model, and random circuit dynamics. We obtain compatible universal properties in each case, indicating that the entangling-disentangling phase transition is generic for projectively measured many-body systems. We discuss the significance of this transition for numerical calculations of quantum observables in many-body systems.

An extension of the Ising spin configurations to continuous functions is used for an exact representation of the random field Ising model’s order parameter in terms of disagreement percolation. This facilitates an extension of the recent analyses of the decay of correlations to positive temperatures, at homogeneous but arbitrarily weak disorder.

We study the solution of the Kardar–Parisi–Zhang (KPZ) equation for the stochastic growth of an interface of height h ( x ,  t ) on the positive half line, equivalently the free energy of the continuum directed polymer in a half space with a wall at x = 0 . The boundary condition ∂ x h ( x , t ) | x = 0 = A corresponds to an attractive wall for A < 0 , and leads to the binding of the polymer to the wall below the critical value A = - 1 / 2 . Here we choose the initial condition h ( x , 0) to be a Brownian motion in x > 0 with drift - ( B + 1 / 2 ) . When A + B → - 1 , the solution is stationary, i.e. h ( · , t ) remains at all times a Brownian motion with the same drift, up to a global height shift h (0,  t ). We show that the distribution of this height shift is invariant under the exchange of parameters A and B . For any A , B > - 1 / 2 , we provide an exact formula characterizing the distribution of h (0,  t ) at any time t , using two methods: the replica Bethe ansatz and a discretization called the log-gamma polymer, for which moment formulae were obtained. We analyze its large time asymptotics for various ranges of parameters A ,  B . In particular, when ( A , B ) → ( - 1 / 2 , - 1 / 2 ) , the critical stationary case, the fluctuations of the interface are governed by a universal distribution akin to the Baik–Rains distribution arising in stationary growth on the full-line. It can be expressed in terms of a simple Fredholm determinant, or equivalently in terms of the Painlevé II transcendent. This provides an analog for the KPZ equation, of some of the results recently obtained by Betea–Ferrari–Occelli in the context of stationary half-space last-passage-percolation. From universality, we expect that limiting distributions found in both models can be shown to coincide.

Our motivation in this paper is twofold. First, we study the geometry of a class of exploration sets, called exit sets , which are naturally associated with a 2D vector-valued Gaussian Free Field :$$\\phi : \\mathbb {Z}^2 \\rightarrow \\mathbb {R}^N, N\\ge 1$$ϕ : Z 2 → R N , N ≥ 1 . We prove that, somewhat surprisingly, these sets are a.s. degenerate as long as$$N\\ge 2$$N ≥ 2 , while they are conjectured to be macroscopic and fractal when$$N=1$$N = 1 . This analysis allows us, when$$N\\ge 2$$N ≥ 2 , to understand the percolation properties of the level sets of$$\\{ \\Vert \\phi (x)\\Vert _{{2}}, xın \\mathbb {Z}^2\\}$$‖ ϕ ( x ) ‖ 2 , x ∈ Z 2 and leads us to our second main motivation in this work: if one projects a spin$$O(N+1)$$O ( N + 1 ) model (the case$$N=2$$N = 2 corresponds to the classical Heisenberg model) down to a spin O ( N ) model, we end up with a spin O ( N ) in a quenched disorder given by random conductances on$$\\mathbb {Z}^2$$Z 2 . Using the exit sets of the N -vector-valued GFF, we obtain a local and geometric description of this random disorder in the limit$$\\beta \\rightarrow ınfty $$β → ∞ . This allows us in particular to revisit a series of celebrated works by Patrascioiu and Seiler (J Stat Phys 69(3):573–595, 1992, Nucl Phys B Proc Suppl 30:184–191, 1993, J Stat Phys 106(3):811–826, 2002) which argued against Polyakov’s prediction that spin$$O(N+1)$$O ( N + 1 ) model is massive at all temperatures as long as$$N\\ge 2$$N ≥ 2 (Polyakov in Phys Lett B 59(1):79–81, 1975). We make part of their arguments rigorous and more importantly we provide the following counter-example: we build ergodic environments of (arbitrary) high conductances with (arbitrary) small and disconnected regions of low conductances in which, despite the predominance of high conductances, the XY model remains massive. Of independent interest, we prove that at high$$\\beta $$β , the fluctuations of a classical Heisenberg model near a north pointing spin are given by a$$N=2$$N = 2 vectorial GFF. This is implicit for example in Polyakov (1975) but we give here the first (non-trivial) rigorous proof. Also, independently of the recent work Dubédat and Falconet (Random clusters in the villain and xy models, arXiv preprint arXiv:2210.03620 , 2022), we show that two-point correlation functions of the spin O ( N ) model can be given in terms of certain percolation events in the cable graph for any$$N\\ge 1$$N ≥ 1 .

In this paper, we investigate the behaviour of statistical physics models on a book with pages that are isomorphic to half-planes. We show that even for models undergoing a continuous phase transition on Z 2 , the phase transition becomes discontinuous as soon as the number of pages is sufficiently large. In particular, we prove that the Ising model on a three pages book has a discontinuous phase transition (if one allows oneself to consider large coupling constants along the line on which pages are glued). Our work confirms predictions in theoretical physics which relied on renormalization group, conformal field theory and numerics (Cardy in J Phys A Math Gen 24(22):L131, 1991; Iglói et al. in J Phys A Math Gen 24(17):L1031, 1991; Stéphan et al. in Phys Rev B 82(12):125455, 2010) some of which were motivated by the analysis of the Renyi entropy of certain quantum spin systems.

Quantum networks have experienced rapid advancements in both theoretical and experimental domains over the last decade, making it increasingly important to understand their large-scale features from the viewpoint of statistical physics. This review paper discusses a fundamental question: how can entanglement be effectively and indirectly (e.g., through intermediate nodes) distributed between distant nodes in an imperfect quantum network, where the connections are only partially entangled and subject to quantum noise? We survey recent studies addressing this issue by drawing exact or approximate mappings to percolation theory, a branch of statistical physics centered on network connectivity. Notably, we show that the classical percolation frameworks do not uniquely define the network’s indirect connectivity. This realization leads to the emergence of an alternative theory called “concurrence percolation”, which uncovers a previously unrecognized quantum advantage that emerges at large scales, suggesting that quantum networks are more resilient than initially assumed within classical percolation contexts, offering refreshing insights into future quantum network design.