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Quenched equals annealed at leading order in the colored SYK model

Razvan Gurau

TL;DR

The paper analyzes a colored SYK model with color-distinct interaction fields and proves that at leading order in the large-$N$ expansion, the quenched and annealed free energies coincide, for any number of colors $D$ and irrespective of fermionic or bosonic statistics or dynamic propagators. The proof leverages edge-colored graph techniques from random tensor models and shows that leading contributions come from degree-zero graphs (melonic for $D\ge 3$ and planar for $D=2$), yielding an all-orders-in-$J$ all-order $1/N$ framework. This establishes a rigorous, replica-free route to equate disorder-averaged thermodynamics at leading order and provides detailed combinatorial counts for the dominant graphs. The results pave the way for systematic study of $1/N$ corrections and suggest avenues to promote the colored model to a dynamical field theory with a melonic large-$N$ phase in vector SYK variants.

Abstract

We consider a colored version of the SYK model, that is we distinguish the $D$ vector fermionic fields involved in the interaction by a color. We obtain the full $1/N$ series of both the quenched and annealed free energies of the model and show that at leading order the two are identical. The results can be used to study systematically the $1/N$ corrections to this leading order behavior.

Quenched equals annealed at leading order in the colored SYK model

TL;DR

The paper analyzes a colored SYK model with color-distinct interaction fields and proves that at leading order in the large- expansion, the quenched and annealed free energies coincide, for any number of colors and irrespective of fermionic or bosonic statistics or dynamic propagators. The proof leverages edge-colored graph techniques from random tensor models and shows that leading contributions come from degree-zero graphs (melonic for and planar for ), yielding an all-orders-in- all-order framework. This establishes a rigorous, replica-free route to equate disorder-averaged thermodynamics at leading order and provides detailed combinatorial counts for the dominant graphs. The results pave the way for systematic study of corrections and suggest avenues to promote the colored model to a dynamical field theory with a melonic large- phase in vector SYK variants.

Abstract

We consider a colored version of the SYK model, that is we distinguish the vector fermionic fields involved in the interaction by a color. We obtain the full series of both the quenched and annealed free energies of the model and show that at leading order the two are identical. The results can be used to study systematically the corrections to this leading order behavior.

Paper Structure

This paper contains 9 sections, 2 theorems, 30 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The colored SYK model
  3. Proof of the main result
  4. Edge colored graphs
  5. The thermal partition function
  6. Integrating the disorder
  7. The leading order graphs
  8. The $D=2$ case.
  9. The $D\ge 3$ case.

Key Result

Theorem 1

RTMreview Let $D\ge 2$. For any connected $(D+1)$--colored graph ${\cal G}$ there exists a non negative integer $\omega({\cal G})$ such that:

Figures (4)

  • Figure 1: Edge $3$-colored graphs.
  • Figure 2: $3$-colored graphs and associated permutations. The permutations $\sigma^{(c)}$ are written in cycle notation, the capital labels designate the vertices and the lower-case labels designate the colors of the edges.
  • Figure 3: The $5$ leading order $(2+1)$-colored graphs with three white vertices. The edges of color $0$ are represented as dashed.
  • Figure 4: Some examples of leading order $(3+1)$-colored graphs with three white vertices.

Theorems & Definitions (5)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof