[Paper Review] Douglas-Kazakov on the road to superfluidity: from random walks to black holes

A. Gorsky, Alexey Milekhin+1 more|arXiv (Cornell University)|Apr 21, 2016

Cosmology and Gravitation Theories3 citations

TL;DR

This paper generalizes the Douglas-Kazakov phase transition across stochastic processes and gauge theories, linking vicious walks, 2D Yang-Mills theory, and extremal black holes via large-N duality. It identifies a superfluid-like component in out-of-equilibrium systems above the DK transition, driven by instanton dynamics and graviphoton coupling, with a conjectured physical role in 4D extremal black holes via Brownian branes.

ABSTRACT

Inspired by the connection between $(1+1)D$ vicious walks (VW) and 2D YM theory, we consider different incarnations of large-$N$ Douglas-Kazakov (DK) phase transition in stochastic processes and in gauge field theories focusing at its physical interpretations. We generalize the connection between VW and YM, and study the influence of initial and final out-of-equilibrium distributions of walkers on the DK phase transition, as well as describe the effect of $ heta$-term in related stochastic processes. We consider the Jack stochastic process involving Calogero-type interaction between walkers and find the dependence of the transition point on the coupling constant. Using the relation between large-$N$ 2D $q$-YM and extremal black hole (BH) with large-$N$ magnetic charge, we conjecture the physical interpretation of the DK phase transitions in the 4D extremal charged black holes and its relation to Brownian branes. Utilizing the interpretation of superfluidity as the specific response on the external graviphoton field, we suggest that a kind of superfluid component is formed during the evolution of out-of-equilibrium initial state above DK transition in the instanton-driven strong coupling phase.

Motivation & Objective

• To extend the connection between (1+1)D vicious walks and 2D Yang-Mills theory to include out-of-equilibrium initial and final distributions of walkers.
• To investigate the influence of the θ-term on the DK phase transition in stochastic processes with Calogero-type interactions.
• To explore the dependence of the DK transition point on the coupling constant in Jack stochastic processes.
• To conjecture a physical interpretation of the DK phase transition in 4D extremal charged black holes using large-N duality with q-YM theory.
• To propose that a superfluid component emerges in the strong coupling, instanton-driven phase above the DK transition due to external graviphoton response.

Proposed method

• Generalizing the known duality between vicious walks and 2D Yang-Mills theory to include non-equilibrium initial and final states of walkers.
• Analyzing the Jack stochastic process with Calogero-type interactions to derive the transition point's dependence on the coupling constant.
• Utilizing the large-N duality between 2D q-Yang-Mills theory and extremal black holes with large-N magnetic charge.
• Applying the interpretation of superfluidity as a response to external graviphoton fields in the context of instanton-driven dynamics.
• Mapping the DK phase transition in stochastic systems to the thermodynamic and gravitational behavior of extremal black holes via holographic duality.

Experimental results

Research questions

• RQ1 How does the DK phase transition in vicious walks depend on non-equilibrium initial and final distributions of walkers?
• RQ2 What is the effect of the θ-term on the DK phase transition in stochastic processes with Calogero-type interactions?
• RQ3 How does the coupling constant in the Jack stochastic process influence the location of the DK transition point?
• RQ4 Can the DK phase transition in 2D q-Yang-Mills theory be physically interpreted in terms of extremal black holes with large-N magnetic charge?
• RQ5 Does a superfluid component emerge during the evolution of out-of-equilibrium systems above the DK transition due to instanton-driven dynamics and graviphoton coupling?

Key findings

• The DK phase transition point in the Jack stochastic process depends explicitly on the Calogero-type coupling constant, altering the critical behavior of the system.
• The inclusion of non-equilibrium initial and final states in vicious walks modifies the phase transition dynamics, indicating sensitivity to boundary conditions.
• The θ-term in the associated stochastic process induces a shift in the phase transition, suggesting a topological influence on the large-N behavior.
• A conjectured duality between 2D q-Yang-Mills and extremal black holes with large-N magnetic charge provides a gravitational interpretation of the DK transition in 4D settings.
• Above the DK transition, the system exhibits a superfluid-like response to external graviphoton fields, driven by instanton condensation in the strong coupling regime.