## Journal of Applied Probability

- J. Appl. Probab.
- Volume 48, Number 3 (2011), 885-891.

### Tree polymers in the infinite volume limit at critical strong disorder

Torrey Johnson and Edward C. Waymire

#### Abstract

The almost-sure existence of a polymer probability in the infinite volume limit is readily obtained under general conditions of weak disorder from standard theory on multiplicative cascades or branching random walks. However, speculations in the case of strong disorder have been mixed. In this note existence of an infinite volume probability is established at critical strong disorder for which one has convergence in probability. Some calculations in support of a specific formula for the almost-sure asymptotic variance of the polymer path under strong disorder are also provided.

#### Article information

**Source**

J. Appl. Probab., Volume 48, Number 3 (2011), 885-891.

**Dates**

First available in Project Euclid: 23 September 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.jap/1316796923

**Digital Object Identifier**

doi:10.1239/jap/1316796923

**Mathematical Reviews number (MathSciNet)**

MR2884824

**Zentralblatt MATH identifier**

1228.60111

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 60G42: Martingales with discrete parameter 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

**Keywords**

Multiplicative cascade T-martingale tree polymer strong disorder

#### Citation

Johnson, Torrey; Waymire, Edward C. Tree polymers in the infinite volume limit at critical strong disorder. J. Appl. Probab. 48 (2011), no. 3, 885--891. doi:10.1239/jap/1316796923. https://projecteuclid.org/euclid.jap/1316796923