LBan-III: Difference between revisions
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(Created page with "= Goal = The goal of this lecture is to present the final lecture on KPZ and directed polymers in finite dimension. We show that for <math>d>2</math>, a "glass transition" occurs. = KPZ: from 1D to the Cayley tree = Much is known about KPZ, but several aspects remain elusive: In <math>d=1</math>, we have <math>\theta=1/3</math> and a glassy regime present at all temperatures. The stationary solution of the KPZ equation describes, at long times, the fluctuations of qu...") |
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The goal of this lecture is to present the final lecture on KPZ and directed polymers in finite dimension. We show that for <math>d>2</math>, a "glass transition" occurs. | The goal of this lecture is to present the final lecture on KPZ and directed polymers in finite dimension. We show that for <math>d>2</math>, a "glass transition" occurs. | ||
= | = Directed polymers from 1D to the Cayley tree = | ||
In <math>d=1</math>, we have <math>\theta=1/3</math> and a glassy regime present at all temperatures. The | * In <math>d=1</math>, we have <math>\theta=1/3</math> and a glassy regime present at all temperatures. The full distribution of <math>E_{\min}</math> for a given <math>x</math> is in the Tracy–Widom family. | ||
In <math>d=\infty</math>, the Cayley tree can be solved exactly, predicting a freezing transition to a 1RSB phase (<math>\theta=0</math>). | * In <math>d=\infty</math>, the Cayley tree can be solved exactly, predicting a freezing transition to a 1RSB phase (<math>\theta=0</math>). | ||
In finite dimensions greater than one, exact solutions are not available. Numerical simulations indicate <math>\theta > 0</math> in <math>d=2</math>, while the case <math>d>2</math> remains particularly intriguing. | * In finite dimensions greater than one, exact solutions are not available. Numerical simulations indicate <math>\theta > 0</math> in <math>d=2</math>, while the case <math>d>2</math> remains particularly intriguing. | ||
= Replica Analysis = | |||
To study disordered systems, we analyze moments of the partition function. From the first lecture, recall that if | |||
Latest revision as of 19:15, 27 August 2025
Goal
The goal of this lecture is to present the final lecture on KPZ and directed polymers in finite dimension. We show that for , a "glass transition" occurs.
Directed polymers from 1D to the Cayley tree
- In , we have and a glassy regime present at all temperatures. The full distribution of for a given is in the Tracy–Widom family.
- In , the Cayley tree can be solved exactly, predicting a freezing transition to a 1RSB phase ().
- In finite dimensions greater than one, exact solutions are not available. Numerical simulations indicate in , while the case remains particularly intriguing.
Replica Analysis
To study disordered systems, we analyze moments of the partition function. From the first lecture, recall that if