arXiv subjects

Victor Ginsburg

Publications and source records attributed to Victor Ginsburg.

3 recordsLinked to original sources

arXiv2026

Last passage percolation in hierarchical environments

Last passage percolation (LPP) is a model of a directed metric and a zero-temperature polymer where the main observable is a directed path evolving in a random environment accruing as energy the sum of the random weights along itself. When the environment has light tails and a fast decay of correlation, the fluctuations of LPP are predicted to be explained by the Kardar-Parisi-Zhang (KPZ) universality theory. However, the KPZ theory is not expected to apply for many natural environments, particularly "critical" ones exhibiting a hierarchical structure often leading to logarithmic correlations. In this article, we initiate a novel study of LPP in such hierarchical environments by investigating two particularly interesting examples. The first is an i.i.d. environment but with a power-law distribution with an inverse quadratic tail decay which is conjectured to be the critical point for the validity of the KPZ scaling relation. The second is the Branching Random Walk which is a hierarchical approximation of the two-dimensional Gaussian Free Field. The second example may be viewed as a high-temperature (weak coupling) directed version of Liouville Quantum Gravity, which is a model of random geometry driven by the exponential of a logarithmically correlated field. Due to the underlying fractal structure, LPP in such environments is expected to exhibit logarithmic correction terms with novel critical exponents. While discussions about such critical models appear in the physics literature, precise predictions about exponents seem to be missing. Developing a framework based on multi-scale analysis, we obtain bounds on such exponents and prove almost optimal concentration results in all dimensions for both models. As a byproduct of our analysis we answer a long-standing question of Martin on necessary and sufficient conditions for the linear growth of the LPP energy in i.i.d. environments.

math.PR↗

arXiv2026

Temperature chaos in directed polymers

Disordered systems such as spin glasses and polymers characteristically exhibit random energy landscapes with many macroscopically separated energetic valleys corresponding to near-ground states. This high complexity renders these systems extremely sensitive to perturbations of external parameters. For instance, the support of associated Gibbs measures may change macroscopically under such perturbations, a phenomenon known as chaos in the literature. In experiments, chaotic phenomena are typically studied via temperature perturbations. In this article, we initiate the rigorous study of temperature-chaotic properties of the continuum directed random polymer (CDRP), a canonical model in the KPZ universality class. The CDRP is driven by white noise and is parametrized by inverse temperature $β$, and is known [Wu '26, Das-Zhu '24] to converge in the zero-temperature limit $β\to \infty$ to the directed landscape constructed in [Dauvergne-Ortmann-Virág '22], the putative universal scaling limit of models in the KPZ universality class. The main result of this article considers the CDRP free energies coupled through the same white noise at a pair of inverse temperatures $(β_1, β_2)$, and shows that they decouple in the limit $β_2 \gg β_1 \gg 1$, converging to a pair of independent directed landscapes. This is the first such "energetic de-correlation across temperatures" result. Our key estimate measures the "pivotality" or "influence" of spatially thin strips in models of last passage percolation. As a byproduct, the proof strategy also allows to show that the directed landscape is a two-dimensional black noise (in the sense of [Tsirelson-Vershik '98]), previously conjectured by Virág. This provides the third known example of a two-dimensional black noise after critical planar percolation [Schramm-Smirnov '11] and the Brownian web [Ellis-Feldheim '16].

math.PR↗

arXiv2024

Pinning, diffusive fluctuations, and Gaussian limits for half-space directed polymer models

Half-space directed polymers in random environments are models of interface growth in the presence of an attractive hard wall. They arise naturally in the study of wetting and entropic repulsion phenomena. In 1985, Kardar predicted a "depinning" phase transition as the attractive force of the wall is weakened. This phase transition has been rigorously established for integrable models of half-space last passage percolation, i.e. half-space directed polymers at zero temperature, in a line of study tracing back to work of Baik--Rains. On the other hand, for integrable positive temperature models, the first rigorous proof of this phase transition has only been obtained very recently through a series of works of Barraquand--Wang, Imamura--Mucciconi--Sasamoto [IMS], Barraquand--Corwin--Das, and Das--Zhu [DZ] on the half-space log-Gamma polymer. In this paper we study a broad class of half-space directed polymer models with minimal assumptions on the random environment. We prove that an attractive force on the wall strong enough to macroscopically increase the free energy induces phenomena characteristic of the subcritical "bound phase," namely the pinning of the polymer to the wall and the diffusive fluctuations and limiting Gaussianity of the free energy. Our arguments are geometric in nature and allow us to analyze the positive temperature and zero temperature models simultaneously. Moreover, given the macroscopic free energy increase proven in [IMS] for the half-space log-Gamma polymer, our arguments can be used to reprove the results of [IMS, DZ] on polymer geometry and free energy fluctuations in the bound phase.

math.PR↗