We propose a multi-scale diagonalization scheme to study disordered one-dimensional chains, in particular the transition between many-body localization (MBL) and the ergodic phase, expected to be governed by resonant spots. Our scheme focuses on the dichotomy MBL versus ETH (eigenstate thermalization hypothesis). We show that a few natural assumptions imply that the system is localized with probability one at criticality. On the ergodic side, delocalization is induced by a quantum avalanche seeded by large ergodic spots, whose size diverges at the transition. On the MBL side, the typical localization length tends to a finite universal value at the transition, but there is a divergent length scale related to the response to an inclusion of large ergodic spots. A mean field approximation analytically illustrates these results and predicts as a power-law distribution for thermal inclusions at criticality.

In a recent work Povolotsky 24 provided a three-parameter family of stochastic particle systems with zero-range interactions in one dimension which are integrable by coordinate Bethe ansatz. Using these results we obtain the corresponding condition for integrability of a class of directed polymer models with random weights on the square lattice. Analyzing the solutions we find, besides known cases, a new two-parameter family of integrable DP model, which we call the Inverse-Beta polymer, and provide its Bethe ansatz solution.PACS numbers: I. INTRODUCTION AND MAIN RESULTS A. overviewThere is considerable recent interest in exact solutions for models in the universality class of the 1D stochastic growth Kardar-Parisi-Zhang equation (KPZ) 1 . Models in the KPZ class share the same large time statistics, also found to be related to the universal statistics of large random matrices 2 . Methods developped in the context of quantum integrability are exploited and broadly extended to solve a variety of 1D stochastic models. The Bethe ansatz solution of the attractive delta Bose gas (the Lieb-Liniger model 3,4 ) was combined with the replica method 5 , to obtain exact solutions for the KPZ equation directly in the continuum and at arbitrary time, for the main classes of initial conditions (droplet, flat, stationary, half-space) 6-15 . The Cole-Hopf mapping h ∼ ln Z is used, where h is the height of the KPZ interface and Z the partition sum of a directed polymer in a random potential (DP). Hence in the continuum, studying KPZ growth is equivalent to studying the DP model, an equilibrium statistical mechanics problem with quenched disorder. The time in KPZ growth becomes the length of the polymer t. The replica Bethe ansatz (RBA) method then allows to calculate the integer moments Z n and, from them, to retrieve the probability distribution function (PDF) of Z. Since the last step is non-rigorous because of the fast growth of these moments, the mathematical community has concentrated on the exact solution of discrete models, which in favorable cases, do not suffer from the moment growth problem. Discrete models, such as the PNG growth model [16][17][18] , the TASEP and ASEP particle transport model 19,20 and discrete DP models 17,21,22 played a pionneering role in unveiling the universal statistics of the KPZ class at large time (the Airy processes). Recently they have been considerably generalized, unveiling a very rich underlying "stochastic integrability" structure 23-33 . Since in suitable limits (e.g. ASEP with weak asymmetry, q-TASEP with q → 1, semi-discrete DP) they converge to the continuum KPZ equation, they also led to some recent rigorous results for KPZ at arbitrary time [35][36][37][38][39] .Besides their interest in relation to KPZ growth, directed polymers are also important in a variety of fields. This includes optimization and glasses 40,41 , vortex lines in superconductors 42 , domain walls in magnets 43 , disordered conductors 44 , Burgers equation in fluid mechanics 45 , exploration-exploitation tradeof...

Abstract. We study the model of a discrete directed polymer (DP) on the square lattice with homogeneous inverse gamma distribution of site random Boltzmann weights, introduced by Seppalainen [1]. The integer moments of the partition sum, Z n , are studied using a transfer matrix formulation, which appears as a generalization of the Lieb-Liniger quantum mechanics of bosons to discrete time and space. In the present case of the inverse gamma distribution the model is integrable in terms of a coordinate Bethe Ansatz, as discovered by Brunet. Using the BrunetBethe eigenstates we obtain an exact expression for the integer moments of Z n for polymers of arbitrary lengths and fixed endpoint positions. Although these moments do not exist for all integer n, we are nevertheless able to construct a generating function which reproduces all existing integer moments, and which takes the form of a Fredholm determinant (FD). This suggests an analytic continuation via a Mellin-Barnes transform and we thereby propose a FD ansatz representation for the probability distribution function (PDF) of Z and its Laplace transform. In the limit of very long DP, this ansatz yields that the distribution of the free energy converges to the Gaussian unitary ensemble (GUE) Tracy-Widom distribution up to a non-trivial average and variance that we calculate. Our asymptotic predictions coincide with a result by Borodin et al. [3] based on a formula obtained by Corwin et al. [2] using the geometric Robinson-Schensted-Knuth (gRSK) correspondence. In addition we obtain the dependence on the endpoint position and the exact elastic coefficient at large time. We argue the equivalence between our formula and the one of Borodin et al. As we discuss, this provides connections between quantum integrability and tropical combinatorics.arXiv:1406.5963v2 [cond-mat.dis-nn]

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Abstract. We study the Brownian force model (BFM), a solvable model of avalanche statistics for an interface, in a general discrete setting. The BFM describes the overdamped motion of elastically coupled particles driven by a parabolic well in independent Brownian force landscapes. Avalanches are defined as the collective jump of the particles in response to an arbitrary monotonous change in the well position (i.e. in the applied force). We derive an exact formula for the joint probability distribution of these jumps. From it we obtain the joint density of local avalanche sizes for stationary driving in the quasi-static limit near the depinning threshold. A saddle-point analysis predicts the spatial shape of avalanches in the limit of large aspect ratios for the continuum version of the model. We then study fluctuations around this saddle point, and obtain the leading corrections to the mean shape, the fluctuations around the mean shape and the shape asymmetry, for finite aspect ratios. Our results are finally confronted to numerical simulations.arXiv:1504.05342v1 [cond-mat.dis-nn]