arXiv subjects

Blaz Mramor

Publications and source records attributed to Blaz Mramor.

8 recordsLinked to original sources

arXiv2026

Less Traffic, Better Outcomes: Competition-Aware Request Dispatch in Real-Time Ad Exchanges

Real-time bidding (RTB) ad exchanges typically forward nearly all incoming requests to demand-side platforms (DSPs), even though only a small fraction receive bids. This over-distribution weakens auction outcomes: DSPs throttle participation under compute and budget constraints, reducing the effective use of limited bidding capacity. We present a competition-aware request dispatch framework that uses distributional bid prediction and probabilistic forwarding to decide whether each request should be sent to each DSP. The system adapts per-DSP thresholds over time through lightweight policy optimization to track non-stationary market conditions. We evaluate the framework through four sequential online experiments on a production platform serving over 20 billion daily requests. A full multi-DSP deployment reduces DSP request volume under the policy by 34.2% while increasing net revenue by 4.6% (p<0.001) in a recent 14-day window after an initial DSP adaptation period. Further analysis highlights strong heterogeneity across traffic segments and reveals that aggregate metrics can be misleading. Segment-level and per-DSP analyses suggest that the policy surfaces comparative advantages among DSPs, improving monetized outcomes without increasing overall request volume.

cs.AI↗

arXiv2017

Minimisers of the Allen-Cahn equation on hyperbolic graphs

We investigate minimal solutions of the Allen-Cahn equation on a Gromov-hyperbolic graph. Under some natural conditions on the graph, we show the existence of non-constant uniformly-bounded minimal solutions with prescribed asymptotic behaviours. For a phase field model on a hyperbolic graph, such solutions describe energy-minimising steady-state phase transitions that converge towards prescribed phases given by the asymptotic directions on the graph.

math.AP↗

arXiv2015

Minimisers of the Allen-Cahn equation and the asymptotic Plateau problem on hyperbolic groups

We investigate the existence of non-constant uniformly-bounded minimal solutions of the Allen-Cahn equation on a Gromov-hyperbolic group. We show that whenever the Laplace term in the Allen-Cahn equation is small enough, there exist minimal solutions satisfying a large class of prescribed asymptotic behaviours. For a phase field model on a hyperbolic group, such solutions describe phase transitions that asymptotically converge towards prescribed phases, given by asymptotic directions. In the spirit of de Giorgi's conjecture, we then fix an asymptotic behaviour and let the Laplace term go to zero. In the limit we obtain a solution to a corresponding asymptotic Plateau problem by $Γ$-convergence.

math.AP↗

arXiv2013

Continuity of the Peierls barrier and robustness of laminations

We study the Peierls barrier for a broad class of monotone variational problems. These problems arise naturally in solid state physics and from Hamiltonian twist maps. We start with the case of a fixed local potential and derive an estimate for the difference of the periodic Peierls barrier and the Peierls barrier of a general rotation number in a given point. A similar estimate was obtained by Mather in the context of twist maps, but our proof is different and applies more generally. It follows from the estimate that the Peierls barrier is continuous at irrational points. Moreover, we show that the Peierls barrier depends continuously on parameters and hence that the property that a monotone variational problem admits a lamination of minimizers for a given rotation number, is open in the C1-topology.

math.DS↗

arXiv2013

The laminations of a crystal near an anti-continuum limit

The anti-continuum limit of a monotone variational recurrence relation consists of a lattice of uncoupled particles in a periodic background. This limit supports many trivial equilibrium states that persist as solutions of the model with small coupling. We investigate when a persisting solution generates a so-called lamination and prove that near the anti-continuum limit the collection of laminations of solutions is homeomorphic to the (N-1)-dimensional simplex, with N the number of distinct local minima of the background potential. This generalizes a result by Baesens and MacKay on twist maps near an anti-integrable limit.

math.DS↗

arXiv2012

A dichotomy theorem for minimizers of monotone recurrence relations

Variational monotone recurrence relations arise in solid state physics as generalizations of the Frenkel-Kontorova model for a ferromagnetic crystal. For such problems, Aubry-Mather theory establishes the existence of "ground states" or "global minimizers" of arbitrary rotation number. A nearest neighbor crystal model is equivalent to a Hamiltonian twist map. In this case, the global minimizers have a special property: they can only cross once. As a nontrivial consequence, every one of them has the Birkhoff property. In crystals with a larger range of interaction and for higher order recurrence relations, the single crossing property does not hold and there can exist global minimizers that are not Birkhoff. In this paper we investigate the crossings of global minimizers. Under a strong twist condition, we prove the following dichotomy: they are either Birkhoff, and thus very regular, or extremely irregular and nonphysical: they then grow exponentially and oscillate. For Birkhoff minimizers, we also prove certain strong ordering properties that are well known for twist maps.

math.DS↗

arXiv2011

Ghost circles in lattice Aubry-Mather theory

Monotone lattice recurrence relations such as the Frenkel-Kontorova lattice, arise in Hamiltonian lattice mechanics as models for fe?rromagnetism and as discretization of elliptic PDEs. Mathematically, they are a multidimensional counterpart of monotone twist maps. They often admit a variational structure, so that the solutions are the stationary points of a formal action function. Classical Aubry-Mather theory establishes the existence of a large collection of solutions of any rotation vector. For irrational rotation vectors this is the well-known Aubry-Mather set. It consists of global minimizers and it may have gaps. In this paper, we study the gradient flow of the formal action function and we prove that every Aubry-Mather set can be interpolated by a continuous gradient-flow invariant family, the so-called "ghost circle". The existence of ghost circles is first proved for rational rotation vectors and Morse action functions. The main technical result is a compactness theorem for ghost circles, based on a parabolic Harnack inequality for the gradient flow, which implies the existence of ghost circles of arbitrary rotation vectors and for arbitrary actions. As a consequence, we can give a simple proof of the fact that when an Aubry-Mather set has a gap, then this gap must be parametrized by minimizers, or contain a non-minimizing solution.

math.DS↗

arXiv2011

On the destruction of minimal foliations

Monotone variational recurrence relations such as the Frenkel-Kontorova lattice, arise in solid state physics, conservative lattice dynamics and as Hamiltonian twist maps. For such recurrence relations, Aubry-Mather theory guarantees the existence of solutions of every rotation number. They are the action minimizers that constitute the Aubry-Mather set. When the rotation number is irrational, the Aubry-Mather set is either connected or a Cantor set. A connected Aubry-Mather set is called a minimal foliation. In the case of twist maps, it describes an invariant circle, while in solid state physics it corresponds to a continuum of ground states. A Cantor Aubry-Mather set is called a minimal lamination. In this paper we prove that when the rotation number of a minimal foliation is either rational or easy to approximate by rational numbers, then the foliation can be destroyed into a lamination by an arbitrarily small smooth perturbation of the recurrence relation. This generalizes a theorem of Mather for twist maps to general recurrence relations.

math.DS↗