Rosso: /* KPZ: from 1D to the Cayley tree */

2025-08-27T17:15:02Z

KPZ: from 1D to the Cayley tree

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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.

	= ~~KPZ:~~ from 1D to the Cayley tree =		= Directed polymers 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 quantities such as <math>E_{\min}[x] - E_{\min}[x']</math>. However, it does not determine the~~ full distribution of <math>E_{\min}</math> for a given <math>x</math>~~. In particular, the origin of~~ the Tracy–Widom ~~distribution remains unclear~~.		* 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

Rosso: 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 $d>2$ , a "glass transition" occurs. = KPZ: from 1D to the Cayley tree = Much is known about KPZ, but several aspects remain elusive: In $d=1$ , we have $\theta=1/3$ and a glassy regime present at all temperatures. The stationary solution of the KPZ equation describes, at long times, the fluctuations of qu..."

2025-08-27T17:10:12Z

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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= 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 quantities such as <math>E_{\min}[x] - E_{\min}[x']</math>. However, it does not determine the full distribution of <math>E_{\min}</math> for a given <math>x</math>. In particular, the origin of the Tracy–Widom distribution remains unclear.

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.

LBan-III - Revision history

Rosso: /* KPZ: from 1D to the Cayley tree */