We show that the largest density of factor of i.i.d. independent sets in the d-regular tree is asymptotically at most (log d)/d as d → ∞. This matches the lower bound given by previous constructions.It follows that the largest independent sets given by local algorithms on random dregular graphs have the same asymptotic density. In contrast, the density of the largest independent sets in these graphs is asymptotically 2(log d)/d.We prove analogous results for Poisson-Galton-Watson trees, which yield bounds for local algorithms on sparse Erdős-Rényi graphs.

We study the multitime distribution in a discrete polynuclear growth model or, equivalently, in directed last‐passage percolation with geometric weights. A formula for the joint multitime distribution function is derived in the discrete setting. It takes the form of a multiple contour integral of a block Fredholm determinant. The asymptotic multitime distribution is then computed by taking the appropriate KPZ‐scaling limit of this formula. This distribution is expected to be universal for models in the Kardar‐Parisi‐Zhang universality class. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

We show that in random K-uniform hypergraphs of constant average degree, for even K ≥ 4, local algorithms defined as factors of i.i.d. can not find nearly maximal cuts when the average degree is sufficiently large. These algorithms have been used frequently to obtain lower bounds for the max-cut problem on random graphs, but it was not known whether they could be successful in finding nearly maximal cuts. This result follows from the fact that the overlap of any two nearly maximal cuts in such hypergraphs does not take values in a certain non-trivial interval -a phenomenon referred to as the overlap gap property -which is proved by comparing diluted models with large average degree with appropriate fully connected spin glass models, and showing the overlap gap property in the latter setting. This paper considers the problem of algorithmically finding nearly optimal spin configurations in the diluted K-spin model. We specifically focus on local algorithms defined as factors of i.i.d., the formal definition of which is provided in Section 2. The diluted K-spin model is also known as the max-cut problem for Kuniform Erdős-Rényi hypergraphs of constant average degree, and also as the random K-XORSAT model. The problem is only interesting for even K and we prove that, for even K ≥ 4, local algorithms fail to find the nearly optimal spin configurations (maximal cuts) once the average degree is large enough.The proof is based on finding a structural constraint for the overlap of any two nearly optimal spin configurations -the overlap gap property -that goes against certain properties of local algorithms. For K = 2, the overlap gap property is not expected to hold, which is why this case is excluded. The structural constraint is derived from recent results on the mean field K-spin spin glass models, in particular, the Parisi formula and the Guerra-Talagrand replica symmetry breaking bound at zero temperature. We begin with a discussion of the model and the notion of algorithms that we use.The K-spin model The set of ±1 spin configurations on N vertices will be denoted by

We study the multi-time distribution in a discrete polynuclear growth model or, equivalently, in directed last-passage percolation with geometric weights. A formula for the joint multi-time distribution function is derived in the discrete setting. It takes the form of a multiple contour integral of a block Fredholm determinant. The asymptotic multi-time distribution is then computed by taking the appropriate KPZ-scaling limit of this formula. The distribution is expected to be universal in the KPZ universality class.

We establish fundamental properties of infinite geodesics and competition interfaces in the directed landscape. We construct infinite geodesics in the directed landscape, establish their uniqueness and coalescence, and define Busemann functions. We show the second class particle in tasep converges under KPZ scaling to a competition interface. Under suitable conditions we show the competition interface has an asymptotic direction, analogous to the speed of a second class particle in tasep, and determine its law. Moreover, we prove the competition interface has an absolutely continuous law on compact sets with respect to infinite geodesics.